62.31.5 problem Ex 5

Internal problem ID [12977]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 54. Change of independent variable. Page 127
Problem number : Ex 5
Date solved : Tuesday, January 28, 2025 at 04:46:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-8 x^{3} y&=4 x^{3} {\mathrm e}^{-x^{2}} \end{align*}

Solution by Maple

Time used: 0.013 (sec). Leaf size: 30

dsolve(x*diff(y(x),x$2)-(2*x^2+1)*diff(y(x),x)-8*x^3*y(x)=4*x^3*exp(-x^2),y(x), singsol=all)
 
\[ y = \frac {\left (-x^{2}+3 c_{1} \right ) {\mathrm e}^{-x^{2}}}{3}+{\mathrm e}^{2 x^{2}} c_{2} \]

Solution by Mathematica

Time used: 0.075 (sec). Leaf size: 38

DSolve[x*D[y[x],{x,2}]-(2*x^2+1)*D[y[x],x]-8*x^3*y[x]==4*x^3*Exp[-x^2],y[x],x,IncludeSingularSolutions -> True]
 
\[ y(x)\to \frac {1}{9} e^{-x^2} \left (-3 x^2+9 c_1 e^{3 x^2}-1+9 c_2\right ) \]