Internal problem ID [10275]
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 53. Change of
dependent variable. Page 125
Problem number: Ex 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-x^{2} y^{\prime }+y x -x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)+x*y(x)=x,y(x), singsol=all)
\[ y \relax (x ) = x c_{2}+\frac {\left (-\left (-x^{3}\right )^{\frac {2}{3}} 3^{\frac {1}{3}} {\mathrm e}^{\frac {x^{3}}{3}}+x^{3} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )\right ) c_{1}}{x^{2}}+1 \]
✓ Solution by Mathematica
Time used: 0.119 (sec). Leaf size: 28
DSolve[y''[x]-x^2*y'[x]+x*y[x]==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{3} c_2 \text {ExpIntegralE}\left (\frac {4}{3},-\frac {x^3}{3}\right )+c_1 x+1 \\ \end{align*}