62.30.1 problem Ex 1

Internal problem ID [12965]
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
Date solved : Tuesday, January 28, 2025 at 04:45:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }+y x&=x \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 52

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

\[ y = 1+\Gamma \left (\frac {2}{3}\right ) \left (-x^{3}\right )^{{1}/{3}} 3^{{2}/{3}} c_{1} -\left (-x^{3}\right )^{{1}/{3}} 3^{{2}/{3}} \Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) c_{1} +3 c_{1} {\mathrm e}^{\frac {x^{3}}{3}}+c_{2} x \]

✓ Solution by Mathematica

Time used: 0.217 (sec). Leaf size: 114

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

\[ y(x)\to x \int _1^x\frac {e^{-\frac {1}{3} K[1]^3} \Gamma \left (-\frac {1}{3},-\frac {1}{3} K[1]^3\right ) K[1] \sqrt [3]{-K[1]^3}}{3 \sqrt [3]{3}}dK[1]-\frac {e^{-\frac {x^3}{3}} \sqrt [3]{-x^3} \left (-1+c_2 e^{\frac {x^3}{3}}\right ) \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )}{3 \sqrt [3]{3}}+c_1 x \]