Internal problem ID [10861]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a x +2 b \right ) y^{\prime }+\left (b a x +b^{2}-a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 55
dsolve(diff(y(x),x$2)+(a*x+2*b)*diff(y(x),x)+(a*b*x-a+b^2)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x \,{\mathrm e}^{-b x}+c_{2} \left ({\mathrm e}^{-b x} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) x +2 \,{\mathrm e}^{-\frac {\left (a x +2 b \right ) x}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.405 (sec). Leaf size: 64
DSolve[y''[x]+(a*x+2*b)*y'[x]+(a*b*x-a+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x e^{-b x} \left (-\sqrt {\frac {\pi }{2}} \sqrt {a} c_2 \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )-\frac {c_2 e^{-\frac {a x^2}{2}}}{x}+c_1\right ) \]