82.54.13 problem Ex. 13

Internal problem ID [19029]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 13
Date solved : Tuesday, January 28, 2025 at 12:46:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x y^{\prime }-y&=f \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 106

dsolve(diff(y(x),x$2)+x*diff(y(x),x)-y(x)=f(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {x \left (\int \left (\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right ) x \,{\mathrm e}^{\frac {x^{2}}{2}}+2\right ) f \left (x \right )d x \right )}{2}+\frac {\left (-\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right ) x -2 \,{\mathrm e}^{-\frac {x^{2}}{2}}\right ) \left (\int x f \left (x \right ) {\mathrm e}^{\frac {x^{2}}{2}}d x \right )}{2}-{\mathrm e}^{-\frac {x^{2}}{2}} c_{1} -\frac {x \left (c_{1} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )-2 c_{2} \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.274 (sec). Leaf size: 148

DSolve[D[y[x],{x,2}]+x*D[y[x],x]-y[x]==f[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (-\sqrt {\frac {\pi }{2}} x \text {erf}\left (\frac {x}{\sqrt {2}}\right )-e^{-\frac {x^2}{2}}\right ) \int _1^xe^{\frac {K[2]^2}{2}} f(K[2]) K[2]dK[2]+x \int _1^x\left (e^{\frac {K[1]^2}{2}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {K[1]}{\sqrt {2}}\right ) K[1] f(K[1])+f(K[1])\right )dK[1]-\sqrt {\frac {\pi }{2}} c_2 x \text {erf}\left (\frac {x}{\sqrt {2}}\right )-c_2 e^{-\frac {x^2}{2}}+c_1 x \]