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Mathematics LibreTexts

5.1E: Homogeneous Linear Equations (Exercises)

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    18316
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    Q5.1.1

    1.

    1. Verify that \(y_1=e^{2x}\) and \(y_2=e^{5x}\) are solutions of \[y''-7y'+10y=0 \tag{A}\] on \((-\infty,\infty)\).
    2. Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=c_1e^{2x}+c_2e^{5x}\) is a solution of (A) on \((-\infty,\infty)\).
    3. Solve the initial value problem \[y''-7y'+10y=0,\quad y(0)=-1,\quad y'(0)=1.\nonumber \]
    4. Solve the initial value problem \[y''-7y'+10y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber \]

    2.

    1. Verify that \(y_1=e^x\cos x\) and \(y_2=e^x\sin x\) are solutions of \[y''-2y'+2y=0 \tag{A}\] on \((-\infty,\infty)\).
    2. 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 (A) on \((-\infty,\infty)\).
    3. Solve the initial value problem \[y''-2y'+2y=0,\quad y(0)=3,\quad y'(0)=-2.\nonumber \]
    4. Solve the initial value problem \[y''-2y'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber \]

    3.

    1. Verify that \(y_1=e^x\) and \(y_2=xe^x\) are solutions of \[y''-2y'+y=0 \tag{A}\] on \((-\infty,\infty)\).
    2. Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=e^x(c_1+c_2x)\) is a solution of (A) on \((-\infty,\infty)\).
    3. Solve the initial value problem \[y''-2y'+y=0,\quad y(0)=7,\quad y'(0)=4.\nonumber \]
    4. Solve the initial value problem \[y''-2y'+y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber \]

    4.

    1. Verify that \(y_1=1/(x-1)\) and \(y_2=1/(x+1)\) are solutions of \[(x^2-1)y''+4xy'+2y=0 \tag{A}\] on \((-\infty,-1)\), \((-1,1)\), and \((1,\infty)\). What is the general solution of (A) on each of these intervals?
    2. Solve the initial value problem \[(x^2-1)y''+4xy'+2y=0,\quad y(0)=-5,\quad y'(0)=1.\nonumber \] What is the interval of validity of the solution?
    3. Graph the solution of the initial value problem.
    4. Verify Abel’s formula for \(y_1\) and \(y_2\), with \(x_0=0\).

    5. Compute the Wronskians of the given sets of functions.

    1. \(\{1, e^{x}\}\)
    2. \(\{e^{x}, e^{x}\sin x\}\)
    3. \(\{x+1, x^{2}+2\}\)
    4. \(\{x^{1/2}, x^{-1/3}\}\)
    5. \(\{\frac{\sin x}{x},\frac{\cos x}{x}\}\)
    6. \(\{x\ln |x|, x^{2}\ln |x|\}\)
    7. \(\{e^{x}\cos\sqrt{x}, e^{x}\sin\sqrt{x}\}\)

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

    \[y''+3(x^2+1)y'-2y=0,\nonumber \]

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

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

    \[(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0,\nonumber \]

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

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

    \[x^2y''+xy'+(x^2-\nu^2)y=0 ,\nonumber \]

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

    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

    \[y''+p(x)y'+q(x)y=0 \tag{A}\]

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

    1. Show that if \(K\) is an arbitrary nonzero constant and \(y_2\) satisfies \[y_1y_2'-y_1'y_2=Ke^{-P(x)} \tag{B}\] on \((a,b)\), then \(y_2\) also satisfies (A) on \((a,b)\), and \(\{y_1,y_2\}\) is a fundamental set of solutions on (A) on \((a,b)\).
    2. Conclude from (a) that if \(y_2=uy_1\) where \(u'=K{e^{-P(x)}\over y_1^2(x)}\), then \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((a,b)\).

    Q5.1.2

    In Exercises 5.1.10-5.1.23 use the method suggested by Exercise 5.1.9 to find a second solution \(y_{2}\) that isn’t a constant multiple of the solution \(y_{1}\). Choose \(K\) conveniently to simplify \(y_{2}\).

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

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

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

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

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

    15. \(x^2y''-(2a-1)xy'+a^2y=0\) (\(a=\) nonzero constant);  \(x>0\); \(y_1=x^a\)

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

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

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

    19. \(4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0\); \(y_1=x^{1/2}\)

    20. \((3x-1)y''-(3x+2)y'-(6x-8)y=0\); \(y_1=e^{2x}\)

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

    22. \((2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0\);\(y_1={1\over x}\)

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

    Q5.1.3

    24. Suppose \(p\) and \(q\) are continuous on an open interval \((a,b)\) and let \(x_0\) be in \((a,b)\). Use Theorem 5.1.1 to show that the only solution of the initial value problem

    \[y''+p(x)y'+q(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0\nonumber \]

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

    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 \((a,b)\).

    1. The initial value problem \[P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \] has more than one solution on \((a,b)\).
    2. The initial value problem \[P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0\nonumber \] has a nontrivial solution on \((a,b)\).

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

    \[y''+p(x)y'+q(x)y=0 \tag{A}\]

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

    \[z_1=\alpha y_1+\beta y_2\quad\text{ and} \quad z_2=\gamma y_1+\delta y_2,\nonumber \]

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

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

    \[y''+p(x)y'+q(x)y=0 \tag{A}\]

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

    \[z_1=\alpha y_1+\beta y_2\quad\text{ and} \quad z_2=\gamma y_1+\delta y_2,\nonumber \]

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

    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 \((a,b)\).

    29. Let

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

    1. Show that the Wronskian of \(\{y_1,y_2\}\) is defined and identically zero on \((-\infty,\infty)\).
    2. Suppose \(a<0<b\). Show that \(\{y_1,y_2\}\) is linearly independent on \((a,b)\).
    3. Use Exercise 5.1.25b to show that these results don’t contradict Theorem 5.1.5, because neither \(y_1\) nor \(y_2\) can be a solution of an equation \[y''+p(x)y'+q(x)y=0\nonumber \] on \((a,b)\) if \(p\) and \(q\) are continuous on \((a,b)\).

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

    \[y''+p(x)y'+q(x)y=0\nonumber \]

    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 \((a,b)\).

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

    \[y''+p(x)y'+q(x)y=0\nonumber \]

    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 \((x_1,x_2)\).

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

    \[y''+p(x)y'+q(x)y=0 \tag{A}\]

    on \((a,b)\) can be written as a linear combination of the twice differentiable functions \(\{y_1,y_2\}\). Use Theorem 5.1.1 to show that \(y_1\) and \(y_2\) are themselves solutions of (A) on \((a,b)\).

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

    \[y''+p_1(x)y'+q_1(x)y=0 \quad \text{and} \quad y''+p_2(x)y'+q_2(x)y=0\nonumber \]

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

    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

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

    can be written as

    \[y''+p(x)y'+q(x)y=0, \tag{A}\]

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

    35. Use the method suggested by Exercise 5.1.34 to find a linear homogeneous equation for which the given functions form a fundamental set of solutions on some interval.

    1. \(e^{x}\cos 2x, e^{x}\sin 2x\)
    2. \(x, e^{2x}\)
    3. \(x, x\ln x\)
    4. \(\cos (\ln x), \sin (\ln x)\)
    5. \(\cosh x, \sinh x\)
    6. \(x^{2}-1, x^{2}+1\)

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

    \[y''+p(x)y'+q(x)y=0 \tag{A}\]

    on \((a,b)\). Show that if \(y\) is a solution of (A) 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 \((a,b)\).

    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

    \[y''+p(x)y'+q(x)y=0 \tag{A}\]

    such that

    \[y_1(x_0)=1, \quad y'_1(x_0)=0\quad \text{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]

    (Theorem 5.1.1 implies that each of these initial value problems has a unique solution on \((a,b)\).)

    1. Show that \(\{y_1,y_2\}\) is linearly independent on \((a,b)\).
    2. Show that an arbitrary solution \(y\) of (A) on \((a,b)\) can be written as \(y=y(x_0)y_1+y'(x_0)y_2\).
    3. Express the solution of the initial value problem \[y''+p(x)y'+q(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \] as a linear combination of \(y_1\) and \(y_2\).

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

    \[y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \text{and} \quad y_2(x_0)=0, \quad y'_2(x_0)=1.\nonumber \]

    Then use Exercise 5.1.37 (c) to write the solution of the initial value problem

    \[y''=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]

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

    39. Let \(x_0\) be an arbitrary real number. Given (Example 5.1.1) 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

    \[y_1(x_0)=1, \quad y'_1(x_0)=0\quad \text{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]

    Then use Exercise 5.1.37 (c) to write the solution of the initial value problem

    \[y''-y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]

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

    40. Let \(x_0\) be an arbitrary real number. Given (Example 5.1.2) that \(\cos\omega x\) and \(\sin\omega x\) are solutions of \(y''+\omega^2y=0\), find solutions of \(y''+\omega^2y=0\) such that

    \[y_1(x_0)=1, \quad y'_1(x_0)=0\quad\text{ and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]

    Then use Exercise 5.1.37 (c) to write the solution of the initial value problem

    \[y''+\omega^2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]

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

    \[\begin{aligned} \cos(A+B)&=\cos A\cos B-\sin A\sin B\\ \sin(A+B)&=\sin A\cos B+\cos A\sin B\end{aligned}\nonumber \]

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

    41. Recall from Exercise 5.1.4 that \(1/(x-1)\) and \(1/(x+1)\) are solutions of

    \[(x^2-1)y''+4xy'+2y=0 \tag{A}\]

    on \((-1,1)\). Find solutions of (A) such that

    \[y_1(0)=1, \quad y'_1(0)=0\quad \text{and} \quad y_2(0)=0,\; y'_2(0)=1.\nonumber \]

    Then use Exercise 5.1.37 (c) to write the solution of initial value problem

    \[(x^2-1)y''+4xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]

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

    42.

    1. Verify that \(y_1=x^2\) and \(y_2=x^3\) satisfy \[x^2y''-4xy'+6y=0 \tag{A}\]  on \((-\infty,\infty)\) and that \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((-\infty,0)\) and \((0,\infty)\).
    2. Let \(a_1\), \(a_2\), \(b_1\), and \(b_2\) be constants. Show that \[y=\left\{\begin{array}{rr} a_1x^2+a_2x^3,&x\ge 0,\\ b_1x^2+b_2x^3,&x<0\phantom{,} \end{array}\right.\nonumber \]  is a solution of (A) on \((-\infty,\infty)\) if and only if \(a_1=b_1\). From this, justify the statement that \(y\) is a solution of (A) on \((-\infty,\infty)\) if and only if \[y=\left\{\begin{array}{rr} c_1x^2+c_2x^3,&x\ge 0,\\ c_1x^2+c_3x^3,&x<0, \end{array}\right.\nonumber \]  where \(c_1\), \(c_2\), and \(c_3\) are arbitrary constants.
    3. For what values of \(k_0\) and \(k_1\) does the initial value problem \[x^2y''-4xy'+6y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]  have a solution? What are the solutions?
    4. Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants, the initial value problem \[x^2y''-4xy'+6y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \tag{B}\]  has infinitely many solutions on \((-\infty,\infty)\). On what interval does (B) have a unique solution?

    43.

    1. Verify that \(y_1=x\) and \(y_2=x^2\) satisfy \[x^2y''-2xy'+2y=0 \tag{A}\]  on \((-\infty,\infty)\) and that \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((-\infty,0)\) and \((0,\infty)\).
    2. Let \(a_1\), \(a_2\), \(b_1\), and \(b_2\) be constants. Show that \[y=\left\{\begin{array}{rr} a_1x+a_2x^2,&x\ge 0,\\ b_1x+b_2x^2,&x<0\phantom{,} \end{array}\right.\nonumber \]  is a solution of (A) 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 (A) on \((-\infty,\infty)\) is \(y=c_1x+c_2x^2\), where \(c_1\) and \(c_2\) are arbitrary constants.
    3. For what values of \(k_0\) and \(k_1\) does the initial value problem \[x^2y''-2xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]  have a solution? What are the solutions?
    4. Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants then the initial value problem \[x^2y''-2xy'+2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]  has a unique solution on \((-\infty,\infty)\).

    44.

    1. Verify that \(y_1=x^3\) and \(y_2=x^4\) satisfy \[x^2y''-6xy'+12y=0 \tag{A}\]  on \((-\infty,\infty)\), and that \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((-\infty,0)\) and \((0,\infty)\).
    2. Show that \(y\) is a solution of (A) on \((-\infty,\infty)\) if and only if \[y=\left\{\begin{array}{rr} a_1x^3+a_2x^4,&x\ge 0,\\ b_1x^3+b_2x^4,&x<0, \end{array}\right.\nonumber \]  where \(a_1\), \(a_2\), \(b_1\), and \(b_2\) are arbitrary constants.
    3. For what values of \(k_0\) and \(k_1\) does the initial value problem \[x^2y''-6xy'+12y=0, \quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]  have a solution? What are the solutions?
    4. Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants then the initial value problem \[x^2y''-6xy'+12y=0, \quad y(x_0)=k_0,\quad y'(x_0)=k_1 \tag{B}\]  has infinitely many solutions on \((-\infty,\infty)\). On what interval does (B) have a unique solution?