3.4E: Exercises
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- 17626
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In Exercises \((3.4E.1)\) to \((3.4E.18)\), find the general solution of the given Euler equation on \((0,\infty)\).
Exercise \(\PageIndex{1}\)
\(x^2y''+7xy'+8y=0\)
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Exercise \(\PageIndex{2}\)
\(x^2y''-7xy'+7y=0\)
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Exercise \(\PageIndex{3}\)
\(x^2y''-xy'+y=0\)
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Exercise \(\PageIndex{4}\)
\(x^2y''+5xy'+4y=0\)
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Exercise \(\PageIndex{5}\)
\(x^2y''+xy'+y=0\)
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Exercise \(\PageIndex{6}\)
\(x^2y''-3xy'+13y=0\)
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Exercise \(\PageIndex{7}\)
\(x^2y''+3xy'-3y=0\)
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Exercise \(\PageIndex{8}\)
\(12x^2y''-5xy''+6y=0\)
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Exercise \(\PageIndex{9}\)
\(4x^2y''+8xy'+y=0\)
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Exercise \(\PageIndex{10}\)
\(3x^2y''-xy'+y=0\)
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Exercise \(\PageIndex{11}\)
\(2x^2y''-3xy'+2y=0\)
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Exercise \(\PageIndex{12}\)
\(x^2y''+3xy'+5y=0\)
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Exercise \(\PageIndex{13}\)
\(9x^2y''+15xy'+y=0\)
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Exercise \(\PageIndex{14}\)
\(x^2y''-xy'+10y=0\)
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Exercise \(\PageIndex{15}\)
\(x^2y''-6y=0\)
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Exercise \(\PageIndex{16}\)
\(2x^2y''+3xy'-y=0\)
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Exercise \(\PageIndex{17}\)
\(x^2y''-3xy'+4y=0\)
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Exercise \(\PageIndex{18}\)
\(2x^2y''+10xy'+9y=0\)
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Exercise \(\PageIndex{19}\)
(a) Adapt the proof of Theorem \((3.4.3)\) to show that \(y=y(x)\) satisfies the Euler equation
\begin{equation}\label{eq:3.4E.1}
ax^2y''+bxy'+cy=0
\end{equation}
on \((-\infty,0)\) if and only if \(Y(t)=y(-e^t)\)
\begin{eqnarray*}
a {d^2Y\over dt^2}+(b-a){dY\over dt}+cY=0.
\end{eqnarray*}
on \((-\infty,\infty)\).
(b) Use part (a) to show that the general solution of \eqref{eq:3.4E.1} on \((-\infty,0)\) is
\begin{eqnarray*}
y&=&c_1|x|^{r_1}+c_2|x|^{r_2}\mbox{ if \(r_1\) and \(r_2\) are distinct real numbers; }\\
y&=&|x|^{r_1}(c_1+c_2\ln|x|)\mbox{ if \(r_1=r_2\); }\\
y&=&|x|^{\lambda}\left[c_1\cos\left(\omega\ln|x|\right)+ c_2\sin\left(\omega\ln|x| \right)\right]\mbox{ if \(r_1,r_2=\lambda\pm i\omega\) with \(\omega>0\)}.
\end{eqnarray*}
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Exercise \(\PageIndex{20}\)
Use reduction of order to show that if
\begin{eqnarray*}
ar(r-1)+br+c=0
\end{eqnarray*}
has a repeated root \(r_1\) then \(y=x^{r_1}(c_1+c_2\ln x)\) is the general solution of
\begin{eqnarray*}
ax^2y''+bxy'+cy=0
\end{eqnarray*}
on \((0,\infty)\).
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Exercise \(\PageIndex{21}\)
A nontrivial solution of
\begin{eqnarray*}
P_0(x)y''+P_1(x)y'+P_2(x)y=0
\end{eqnarray*}
is said to be \( \textcolor{blue}{\mbox{oscillatory}} \) on an interval \((a,b)\) if it has infinitely many zeros on \((a,b)\). Otherwise \(y\) is said to be \( \textcolor{blue}{\mbox{nonoscillatory}} \) on \((a,b)\). Show that the equation
\begin{eqnarray*}
x^2y''+ky=0 \quad (k=\; \mbox{constant})
\end{eqnarray*}
has oscillatory solutions on \((0,\infty)\) if and only if \(k>1/4\).
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Exercise \(\PageIndex{22}\)
In Example \((3.4.2)\) we saw that \(x_0=1\) and \(x_0=-1\) are regular singular points of Legendre's equation
\begin{equation}\label{eq:3.4E.2}
(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0.
\end{equation}
(a) Introduce the new variables \(t=x-1\) and \(Y(t)=y(t+1)\), and show that \(y\) is a solution of \eqref{eq:3.4E.2} if and only if \(Y\) is a solution of
\begin{eqnarray*}
t(2+t){d^2Y\over dt^2}+2(1+t){dY\over dt}-\alpha(\alpha+1)Y=0,
\end{eqnarray*}
which has a regular singular point at \(t_0=0\).
(b) Introduce the new variables \(t=x+1\) and \(Y(t)=y(t-1)\), and show that \(y\) is a solution of \eqref{eq:3.4E.2} if and only if \(Y\( is a solution of
\begin{eqnarray*}
t(2-t){d^2Y\over dt^2}+2(1-t){dY\over dt}+\alpha(\alpha+1)Y=0,
\end{eqnarray*}
which has a regular singular point at \(t_0=0\).
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Exercise \(\PageIndex{23}\)
Let \(P_0,P_1\), and \(P_2\) be polynomials with no common factor, and suppose \(x_0\ne0\) is a singular point of
\begin{equation}\label{eq:3.4E.3}
P_0(x)y''+P_1(x)y'+P_2(x)y=0.
\end{equation}
Let \(t=x-x_0\) and \(Y(t)=y(t+x_0)\).
(a) Show that \(y\) is a solution of \eqref{eq:3.4E.3} if and only if \(Y\) is a solution of
\begin{equation}\label{eq:3.4E.4}
R_0(t){d^2Y\over dt^2}+R_1(t){dY\over dt}+R_2(t)Y=0.
\end{equation}
where
\begin{eqnarray*}
R_i(t)=P_i(t+x_0),\quad i=0,1,2.
\end{eqnarray*}
(b) Show that \(R_0\), \(R_1\), and \(R_2\) are polynomials in \(t\) with no common factors, and \(R_0(0)=0\); thus, \(t_0=0\) is a singular point of \eqref{eq:3.4E.4}.
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