82.49.2 problem Ex. 2

Internal problem ID [19004]
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 page 111
Problem number : Ex. 2
Date solved : Tuesday, January 28, 2025 at 12:45:11 PM
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)
 

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

✓ Solution by Mathematica

Time used: 0.246 (sec). Leaf size: 42

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

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