29.6.16 problem 162

Internal problem ID [4762]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 6
Problem number : 162
Date solved : Monday, January 27, 2025 at 09:36:41 AM
CAS classification : [_linear]

\begin{align*} x y^{\prime }+x +\left (-a \,x^{2}+2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 57

dsolve(x*diff(y(x),x)+x+(-a*x^2+2)*y(x) = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.102 (sec). Leaf size: 70

DSolve[x D[y[x],x]+x+(2-a x^2)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {-\frac {\sqrt {2 \pi } e^{\frac {a x^2}{2}} \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )}{a^{3/2}}+2 c_1 e^{\frac {a x^2}{2}}+\frac {2 x}{a}}{2 x^2} \]