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Mathematics LibreTexts

2.1E: Exercises

  • Page ID
    17281
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    Exercise \(\PageIndex{1}\)

    (a) Verify that \(y_1=e^{2x}\) and \(y_2=e^{5x}\) are solutions of

    \begin{equation}\label{eq:2.1.1}
    y''-7y'+10y=0
    \end{equation}

    on \((-\infty,\infty)\).

    (b) Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=c_1e^{2x}+c_2e^{5x}\) is a solution of \eqref{eq:2.1.1} on \((-\infty,\infty)\).

    (c) Solve the initial value problem

    \begin{eqnarray*}
    y''-7y'+10y=0,\quad y(0)=-1,\quad y'(0)=1.
    \end{eqnarray*}

    (d) Solve the initial value problem

    \begin{eqnarray*}
    y''-7y'+10y=0,\quad y(0)=k_0,\quad y'(0)=k_1.
    \end{eqnarray*}

    Answer

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    Exercise \(\PageIndex{2}\)

    (a) Verify that \(y_1=e^x\cos x\) and \(y_2=e^x\sin x\) are solutions of

    \begin{equation}\label{eq:2.1.2}
    y''-2y'+2y=0
    \end{equation}

    on \((-\infty,\infty)\).

    (b) Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=c_1e^x\cos x+c_2e^x\sin x\) is a solution of \eqref{eq:2.1.2} on \((-\infty,\infty)\).

    (c) Solve the initial value problem

    \begin{eqnarray*}
    y''-2y'+2y=0,\quad y(0)=3,\quad y'(0)=-2.
    \end{eqnarray*}

    (d) Solve the initial value problem

    \begin{eqnarray*}
    y''-2y'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1.
    \end{eqnarray*}

    Answer

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    Exercise \(\PageIndex{3}\)

    (a) Verify that \(y_1=e^x\) and \(y_2=xe^x\) are solutions of

    \begin{equation}\label{eq:2.1.3}
    y''-2y'+y=0
    \end{equation}

    on \((-\infty,\infty)\).

    (b) Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=e^x(c_1+c_2x)\) is a solution of \eqref{eq:2.1.3} on \((-\infty,\infty)\).

    (c) Solve the initial value problem

    \begin{eqnarray*}
    y''-2y'+y=0,\quad y(0)=7,\quad y'(0)=4.
    \end{eqnarray*}

    (d) Solve the initial value problem

    \begin{eqnarray*}
    y''-2y'+y=0,\quad y(0)=k_0,\quad y'(0)=k_1.
    \end{eqnarray*}

    Answer

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    Exercise \(\PageIndex{4}\)

    (a) Verify that \(y_1=1/(x-1)\) and \(y_2=1/(x+1)\) are solutions of

    \begin{equation}\label{eq:2.1.4}
    (x^2-1)y''+4xy'+2y=0
    \end{equation}

    on \((-\infty,-1)\), \((-1,1)\), and \((1,\infty)\). What is the general solution of \eqref{eq:2.1.4} on each of these intervals?

    (b) Solve the initial value problem

    \begin{eqnarray*}
    (x^2-1)y''+4xy'+2y=0,\quad y(0)=-5,\quad y'(0)=1.
    \end{eqnarray*}

    What is the interval of validity of the solution?

    (c) Graph the solution of the initial value problem.

    (d) Verify Abel's formula for \(y_1\) and \(y_2\), with \(x_0=0\).

    Answer

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    Exercise \(\PageIndex{5}\)

    Compute the Wronskians of the given sets of functions.

    (a) \(\{1, e^x\}\)

    (b)\(\{e^x, e^x \sin x\}\)

    (c) \(\{x+1, x^2+2\}\)

    (d) \(\{ x^{1/2}, x^{-1/3}\}\)

    (e) \(\left\{\displaystyle \frac{\sin x}{x}, \frac{\cos x}{x}\right\}\)

    (f) \(\{ x \ln|x|, x^2\ln|x|\}\)

    (g) \(\{e^x\cos\sqrt x, e^x\sin\sqrt x\}\)

    Answer

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    Exercise \(\PageIndex{6}\)

    Find the Wronskian of a given set \(\{y_1,y_2\}\) of solutions of

    \begin{eqnarray*}
    y''+3(x^2+1)y'-2y=0,
    \end{eqnarray*}

    given that \(W(\pi)=0\).

    Answer

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    Exercise \(\PageIndex{7}\)

    Find the Wronskian of a given set \(\{y_1,y_2\}\) of solutions of

    \begin{eqnarray*}
    (1-x^2)y''-2xy'+\alpha(\alpha+1)y=0,
    \end{eqnarray*}

    given that \(W(0)=1\). (This is Legendre's equation.)

    Answer

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    Exercise \(\PageIndex{8}\)

    Find the Wronskian of a given set \(\{y_1,y_2\}\) of solutions of

    \begin{eqnarray*}
    x^2y''+xy'+(x^2-\nu^2)y=0 ,
    \end{eqnarray*}

    given that \(W(1)=1\). (This is Bessel's equation.)

    Answer

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    Exercise \(\PageIndex{9}\)

    (This exercise shows that if you know one nontrivial solution of \(y''+p(x)y'+q(x)y=0\), you can use Abel's formula to find another.)

    Suppose \(p\) and \(q\) are continuous and \(y_1\) is a solution of

    \begin{equation}\label{eq:2.1.5}
    y''+p(x)y'+q(x)y=0
    \end{equation}

    that has no zeros on \((a,b)\). Let \(P(x)=\int p(x)\,dx\) be any antiderivative of \(p\) on \)(a,b)\).

    (a) Show that if \(K\) is an arbitrary nonzero constant and \(y_2\) satisfies

    \begin{equation}\label{eq:2.1.6}
    y_1y_2'-y_1'y_2=Ke^{-P(x)}
    \end{equation}

    on \((a,b)\), then \(y_2\) also satisfies \eqref{eq:2.1.5} on \((a,b)\), and \(\{y_1,y_2\}\) is a fundmental set of solutions on \eqref{eq:2.1.5} on \((a,b)\).

    (b) Conclude from (a) that if \(y_2=uy_1\) where \(u'=K\displaystyle{e^{-P(x)}\over y_1^2(x)}\), then \(\{y_1,y_2\}\) is a fundamental set of solutions of \eqref{eq:2.1.5} on \((a,b)\).

    Answer

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    In Exercises \((2.1E.10)\) to \((2.1E.23)\) use the method suggested by Exercise \((2.1E.9)\) to find a second solution \(y_2\) that isn't a constant multiple of the solution \(y_1\). Choose \(K\) conveniently to simplify \(y_2\).

    Exercise \(\PageIndex{10}\)

    \(y''-2y'-3y=0; \quad y_1=e^{3x}\)

    Answer

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    Exercise \(\PageIndex{11}\)

    \(y''-6y'+9y=0; \quad y_1=e^{3x}\)

    Answer

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    Exercise \(\PageIndex{12}\)

    \(y''-2ay'+a^2y=0; \mbox{(\(a=\) constant)}; \quad y_1=e^{ax}\)

    Answer

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    Exercise \(\PageIndex{13}\)

    \(x^2y''+xy'-y=0; \quad y_1=x\)

    Answer

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    Exercise \(\PageIndex{14}\)

    \(x^2y''-xy'+y=0; \quad y_1=x\)

    Answer

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    Exercise \(\PageIndex{15}\)

    \(x^2y''-(2a-1)xy'+a^2y=0; \mbox{(\(a=\) nonzero constant), \(x>0\);} \quad y_1=x^a\)

    Answer

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    Exercise \(\PageIndex{16}\)

    \(4x^2y''-4xy'+(3-16x^2)y=0; \quad y_1=x^{1/2}e^{2x}\)

    Answer

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    Exercise \(\PageIndex{17}\)

    \((x-1)y''-xy'+y=0; \quad y_1=e^x\)

    Answer

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    Exercise \(\PageIndex{18}\)

    \(x^2y''-2xy'+(x^2+2)y=0; \quad y_1=x\cos x\)

    Answer

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    Exercise \(\PageIndex{19}\)

    \(4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0; \quad y_1=x^{1/2}\)

    Answer

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    Exercise \(\PageIndex{20}\)

    \((3x-1)y''-(3x+2)y'-(6x-8)y=0; \quad y_1=e^{2x}\)

    Answer

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    Exercise \(\PageIndex{21}\)

    \((x^2-4)y''+4xy'+2y=0; \quad y_1=\displaystyle{1\over x-2}\)

    Answer

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    Exercise \(\PageIndex{22}\)

    \((2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0;\quad y_1=\displaystyle{1\over x}\)

    Answer

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    Exercise \(\PageIndex{23}\)

    \((x^2-2x)y''+(2-x^2)y'+(2x-2)y=0;\quad y_1=e^x\)

    Answer

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    Exercise \(\PageIndex{24}\)

    Suppose \(p\) and \(q\) are continuous on an open interval \((a,b)\) and let \(x_0\) be in \((a,b)\). Use Theorem \((2.1.1)\) to show that the only solution of the initial value problem

    \begin{eqnarray*}
    y''+p(x)y'+q(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0
    \end{eqnarray*}

    on \((a,b)\) is the trivial solution \(y\equiv0\).

    Answer

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    Exercise \(\PageIndex{25}\)

    Suppose \(P_0\), \(P_1\), and \(P_2\) are continuous on \((a,b)\) and let \(x_0\) be in \((a,b)\). Show that if either of the following statements is true then \(P_0(x)=0\) for some \(x\) in \((a,b)\).

    (a) The initial value problem

    \begin{eqnarray*}
    P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1
    \end{eqnarray*}

    has more than one solution on \((a,b)\).

    (b) The initial value problem

    \begin{eqnarray*}
    P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0
    \end{eqnarray*}

    has a nontrivial solution on \((a,b)\).

    Answer

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    Exercise \(\PageIndex{26}\)

    Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(y_1\) and \(y_2\) are solutions of

    \begin{equation}\label{eq:2.1.7}
    y''+p(x)y'+q(x)y=0
    \end{equation}

    on \((a,b)\). Let

    \begin{eqnarray*}
    z_1=\alpha y_1+\beta y_2\quad \mbox{and} \quad z_2=\gamma y_1+\delta y_2,
    \end{eqnarray*}

    where \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are constants. Show that if \(\{z_1,z_2\}\) is a fundamental set of solutions of \eqref{eq:2.1.7} on \((a,b)\) then so is \(\{y_1,y_2\}\).

    Answer

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    Exercise \(\PageIndex{27}\)

    Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of

    \begin{equation}\label{eq:2.1.8}
    y''+p(x)y'+q(x)y=0
    \end{equation}

    on \((a,b)\). Let

    \begin{eqnarray*}
    z_1=\alpha y_1+\beta y_2\quad \mbox{and} \quad z_2=\gamma y_1+\delta y_2,
    \end{eqnarray*}

    where \(\alpha,\beta,\gamma\), and \(\delta\) are constants. Show that \(\{z_1,z_2\}\) is a fundamental set of solutions of \eqref{eq:2.1.8} on \((a,b)\) if and only if \( \alpha\gamma-\beta\delta\ne0\).

    Answer

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    Exercise \(\PageIndex{28}\)

    Suppose \(y_1\) is differentiable on an interval \((a,b)\) and \(y_2=ky_1\), where \(k\) is a constant. Show that the Wronskian of \(\{y_1,y_2\}\) is identically zero on \((a,b)\).

    Answer

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    Exercise \(\PageIndex{29}\)

    Let

    \begin{eqnarray*}
    y_1=x^3 \quad \mbox{ and } \quad y_2=\left\{\begin{array}{rl}
    x^3,&x\ge 0,\\ -x^3,&x<0.\end{array}\right.
    \end{eqnarray*}

    (a) Show that the Wronskian of \(\{y_1,y_2\}\) is defined and identically zero on \((-\infty,\infty)\).

    (b) Suppose \(a<0<b\). Show that \(\{y_1,y_2\}\) is linearly independent on \((a,b)\).

    (c) Use Exercise \((2.1E.25)\) part (b) to show that these results don't contradict Theorem \((2.1.5)\), because neither \(y_1\) nor \(y_2\) can be a solution of an equation

    \begin{eqnarray*}
    y''+p(x)y'+q(x)y=0
    \end{eqnarray*}

    on \((a,b)\) if \(p\) and \(q\) are continuous on \((a,b)\).

    Answer

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    Exercise \(\PageIndex{30}\)

    Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a set of solutions of

    \begin{eqnarray*}
    y''+p(x)y'+q(x)y=0
    \end{eqnarray*}

    on \((a,b)\) such that either \(y_1(x_0)=y_2(x_0)=0\) or \(y_1'(x_0)=y_2'(x_0)=0\) for some \(x_0\) in \((a,b)\). Show that \(\{y_1,y_2\}\) is linearly dependent on \((a,b)\).

    Answer

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    Exercise \(\PageIndex{31}\)

    Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of

    \begin{eqnarray*}
    y''+p(x)y'+q(x)y=0
    \end{eqnarray*}

    on \((a,b)\). Show that if \(y_1(x_1)=y_1(x_2)=0\), where \(a<x_1<x_2<b\), then \(y_2(x)=0\) for some \(x\) in \((x_1,x_2)\).

    Hint: Show that if \(y_2\) has no zeros in \((x_1,x_2)\), then \(y_1/y_2\) is either strictly increasing or strictly decreasing on \((x_1,x_2)\), and deduce a contradiction.

    Answer

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    Exercise \(\PageIndex{32}\)

    Suppose \(p\) and \(q\) are continuous on \((a,b)\) and every solution of

    \begin{equation}\label{eq:2.1.9}
    y''+p(x)y'+q(x)y=0
    \end{equation}

    on \((a,b)\) can be written as a linear combination of the twice differentiable functions \(\{y_1,y_2\}\). Use Theorem \((2.1.1)\) to show that \(y_1\) and \(y_2\) are themselves solutions of \eqref{eq:2.1.9} on \((a,b)\).

    Answer

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    Exercise \(\PageIndex{33}\)

    Suppose \(p_1\), \(p_2\), \(q_1\), and \(q_2\) are continuous on \((a,b)\) and the equations

    \begin{eqnarray*}
    y''+p_1(x)y'+q_1(x)y=0 \quad \mbox{and} \quad y''+p_2(x)y'+q_2(x)y=0
    \end{eqnarray*}

    have the same solutions on \((a,b)\). Show that \(p_1=p_2\) and \(q_1=q_2\) on \((a,b)\).

    Hint: Use Abel's formula.

    Answer

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    Exercise \(\PageIndex{34}\)

    (For this exercise you have to know about \(3\times 3\) determinants.)

    Show that if \(y_1\) and \(y_2\) are twice continuously differentiable on \((a,b)\) and the Wronskian \(W\) of \(\{y_1,y_2\}\) has no zeros in \((a,b)\) then the equation

    \begin{eqnarray*}
    \frac{1}{W} \left| \begin{array}{ccc}
    y & y_1 & y_2 \\
    y' & y'_1 & y'_2 \\
    y'' & y_1'' & y_2''
    \end{array} \right|=0
    \end{eqnarray*}

    can be written as

    \begin{equation}\label{eq:2.1.10}
    y''+p(x)y'+q(x)y=0,
    \end{equation}

    where \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of \eqref{eq:2.1.10} on \((a,b)\).

    Hint: Expand the determinant by cofactors of its first column.

    Answer

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    Exercise \(\PageIndex{35}\)

    Use the method suggested by Exercise \((2.1E.34)\) to find a linear homogeneous equation for which the given functions form a fundamental set of solutions on some interval.

    (a) \(e^x \cos 2x, \quad e^x \sin 2x\)

    (b) \(x, \quad e^{2x}\)

    (c) \(x, \quad x \ln x\)

    (d) \(\cos (\ln x), \quad \sin (\ln x)\)

    (e) \(\cosh x, \quad \sinh x\)

    (f) \(x^2-1, \quad x^2+1\)

    Answer

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    Exercise \(\PageIndex{36}\)

    Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of

    \begin{equation}\label{eq:2.1E.11}
    y''+p(x)y'+q(x)y=0
    \end{equation}

    on \((a,b)\). Show that if \(y\) is a solution of \eqref{eq:2.1E.11} on \((a,b)\), there's exactly one way to choose \(c_1\) and \(c_2\) so that \(y=c_1y_1+c_2y_2\) on \((a,b)\).

    Answer

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    Exercise \(\PageIndex{37}\)

    Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(x_0\) is in \((a,b)\). Let \(y_1\) and \(y_2\) be the solutions of

    \begin{equation}\label{eq:2.1E.12}
    y''+p(x)y'+q(x)y=0
    \end{equation}

    such that

    \begin{eqnarray*}
    y_1(x_0)=1, \quad y'_1(x_0)=0\quad \mbox{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.
    \end{eqnarray*}

    (Theorem \((2.1.1)\) implies that each of these initial value problems has a unique solution on \((a,b)\).)

    (a) Show that \(\{y_1,y_2\}\) is linearly independent on \((a,b)\).

    (b) Show that an arbitrary solution \(y\) of \eqref{eq:2.1E.12} on \(a,b)\) can be written as \(y=y(x_0)y_1+y'(x_0)y_2\).

    (c) Express the solution of the initial value problem

    \begin{eqnarray*}
    y''+p(x)y'+q(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1
    \end{eqnarray*}

    as a linear combination of \(y_1\) and \(y_2\).

    Answer

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    Exercise \(\PageIndex{38}\)

    Find solutions \(y_1\) and \(y_2\) of the equation \(y''=0\) that satisfy the initial conditions

    \begin{eqnarray*}
    y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \mbox{ and } \quad y_2(x_0)=0, \quad y'_2(x_0)=1.
    \end{eqnarray*}

    Then use Exercise \((2.1E.37)\) (c) to write the solution of the initial value problem

    \begin{eqnarray*}
    y''=0,\quad y(0)=k_0,\quad y'(0)=k_1
    \end{eqnarray*}

    as a linear combination of \(y_1\) and \(y_2\).

    Answer

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    Exercise \(\PageIndex{39}\)

    Let \(x_0\) be an arbitrary real number. Given Example \((2.1.1)\) that \(e^x\) and \(e^{-x}\) are solutions of \(y''-y=0\), find solutions \(y_1\) and \(y_2\) of \(y''-y=0\) such that

    \begin{eqnarray*}
    y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \mbox{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.
    \end{eqnarray*}

    Then use Exercise \((2.1E.37)\) (c) to write the solution of the initial value problem

    \begin{eqnarray*}
    y''-y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1
    \end{eqnarray*}

    as a linear combination of \(y_1\) and \(y_2\).

    Answer

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    Exercise \(\PageIndex{40}\)

    Let \(x_0\) be an arbitrary real number. Given Example \((2.1.2)\) that \(\cos\omega x\) and \(\sin\omega x\) are solutions of \(y''+\omega^2y=0\), find solutions of \(y''+\omega^2y=0$\) such that

    \begin{eqnarray*}
    y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \mbox{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.
    \end{eqnarray*}

    Then use Exercise \((2.1E.37)\) (c) to write the solution of the initial value problem

    \begin{eqnarray*}
    y''+\omega^2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1
    \end{eqnarray*}

    as a linear combination of \(y_1\) and \(y_2\). Use the identities
    \begin{eqnarray*}
    \cos(A+B)&=&\cos A\cos B-\sin A\sin B\\
    \sin(A+B)&=&\sin A\cos B+\cos A\sin B
    \end{eqnarray*}

    to simplify your expressions for \(y_1\), \(y_2\), and \(y\).

    Answer

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    Exercise \(\PageIndex{41}\)

    Recall from Exercise \((2.1E.4)\) that \(1/(x-1)\) and \(1/(x+1)\) are solutions of

    \begin{equation}\label{eq:2.1E.13}
    (x^2-1)y''+4xy'+2y=0
    \end{equation}

    on \((-1,1)\). Find solutions of \eqref{eq:2.1E.13} such that

    \begin{eqnarray*}
    y_1(0)=1, \quad y'_1(0)=0 \quad \mbox{and} \quad y_2(0)=0,\; y'_2(0)=1.
    \end{eqnarray*}

    Then use Exercise \((2.1E.37)\) (c) to write the solution of initial value problem

    \begin{eqnarray*}
    (x^2-1)y''+4xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1
    \end{eqnarray*}

    as a linear combination of \(y_1\) and \(y_2\).

    Answer

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    Exercise \(\PageIndex{42}\)

    (a) Verify that \(y_1=x^2\) and \(y_2=x^3\) satisfy

    \begin{equation}\label{eq:2.1E.14}
    x^2y''-4xy'+6y=0
    \end{equation}

    on \((-\infty,\infty)\) and that \(\{y_1,y_2\}\) is a fundamental set of solutions of \eqref{eq:2.1E.14} on \((-\infty,0)\) and \((0,\infty)\).

    (b) Let \(a_1\), \(a_2\), \(b_1\), and \(b_2\) be constants. Show that

    \begin{eqnarray*}
    y=\left\{\begin{array}{rr}
    a_1x^2+a_2x^3,&x\ge 0,\\
    b_1x^2+b_2x^3,&x<0\phantom{,}
    \end{array}\right.
    \end{eqnarray*}

    is a solution of \eqref{eq:2.1E.14} on \((-\infty,\infty)\) if and only if \(a_1=b_1\). From this, justify the statement that \(y\) is a solution of \eqref{eq:2.1E.14} on \((-\infty,\infty)\) if and only if

    \begin{eqnarray*}
    y=\left\{\begin{array}{rr}
    c_1x^2+c_2x^3,&x\ge 0,\\
    c_1x^2+c_3x^3,&x<0,
    \end{array}\right.
    \end{eqnarray*}

    where \(c_1\), \(c_2\), and \(c_3\) are arbitrary constants.

    (c) For what values of \(k_0\) and \(k_1\) does the initial value problem

    \begin{eqnarray*}
    x^2y''-4xy'+6y=0,\quad y(0)=k_0,\quad y'(0)=k_1
    \end{eqnarray*}

    have a solution? What are the solutions?

    (d) Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants, the initial value problem

    \begin{equation}\label{eq:2.1E.15}
    x^2y''-4xy'+6y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1
    \end{equation}

    has infinitely many solutions on \((-\infty,\infty)\). On what interval does \eqref{eq:2.1E.15} have a unique solution?

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{43}\)

    (a) Verify that \(y_1=x\) and \(y_2=x^2\) satisfy

    \begin{equation}\label{eq:2.1E.16}
    x^2y''-2xy'+2y=0
    \end{equation}

    on \((-\infty,\infty)\) and that \(\{y_1,y_2\}\) is a fundamental set of solutions of \eqref{eq:2.1E.16} on \((-\infty,0)\) and \((0,\infty)\).

    (b) Let \(a_1\), \(a_2\), \(b_1\), and \(b_2\) be constants. Show that

    \begin{eqnarray*}
    y=\left\{\begin{array}{rr}
    a_1x+a_2x^2,&x\ge 0,\\
    b_1x+b_2x^2,&x<0\phantom{,}
    \end{array}\right.
    \end{eqnarray*}

    is a solution of \eqref{eq:2.1E.16} on \((-\infty,\infty)\) if and only if \(a_1=b_1\) and \(a_2=b_2\). From this, justify the statement that the general solution of \eqref{eq:2.1E.16} on \((-\infty,\infty)\) is \(y=c_1x+c_2x^2\), where \(c_1\) and \(c_2\) are arbitrary constants.

    (c) For what values of \(k_0\) and \(k_1\) does the initial value problem

    \begin{eqnarray*}
    x^2y''-2xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1
    \end{eqnarray*}

    have a solution? What are the solutions?

    (d) Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants then the initial value problem

    \begin{eqnarray*}
    x^2y''-2xy'+2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1
    \end{eqnarray*}

    has a unique solution on \((-\infty,\infty)\).

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{44}\)

    (a) Verify that \(y_1=x^3\) and \(y_2=x^4\) satisfy

    \begin{equation}\label{eq:2.1E.17}
    x^2y''-6xy'+12y=0
    \end{equation}

    on \((-\infty,\infty)\), and that \(\{y_1,y_2\}\) is a fundamental set of solutions of \eqref{eq:2.1E.17} on \((-\infty,0)\) and \((0,\infty)\).

    (b) Show that \(y\) is a solution of \eqref{eq:2.1E.17} on \((-\infty,\infty)\) if and only if

    \begin{eqnarray*}
    y=\left\{\begin{array}{rr}
    a_1x^3+a_2x^4,&x\ge 0,\\
    b_1x^3+b_2x^4,&x<0,
    \end{array}\right.
    \end{eqnarray*}

    where \(a_1\), \(a_2\), \(b_1\), and \(b_2\) are arbitrary constants.

    (c) For what values of \(k_0\) and \(k_1\) does the initial value problem

    \begin{eqnarray*}
    x^2y''-6xy'+12y=0, \quad y(0)=k_0,\quad y'(0)=k_1
    \end{eqnarray*}

    have a solution? What are the solutions?

    (d) Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants then the initial value problem

    \begin{equation}\label{eq:2.1E.18}
    x^2y''-6xy'+12y=0, \quad y(x_0)=k_0,\quad y'(x_0)=k_1
    \end{equation}

    has infinitely many solutions on \((-\infty,\infty)\). On what interval does \eqref{eq:2.1E.18} have a unique solution?

    Answer

    Add texts here. Do not delete this text first.