Internal problem ID [10287]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-8 x^{3} y-4 x^{3} {\mathrm e}^{-x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
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)
\[ y \relax (x ) = {\mathrm e}^{2 x^{2}} c_{2}+{\mathrm e}^{-x^{2}} c_{1}-\frac {x^{2} {\mathrm e}^{-x^{2}}}{3} \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 38
DSolve[x*y''[x]-(2*x^2+1)*y'[x]-8*x^3*y[x]==4*x^3*Exp[-x^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 ) \\ \end{align*}