Internal problem ID [10864]
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: 30.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+2 y^{\prime } \left (a x +b \right )+\left (a^{2} x^{2}+2 b a x +c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 55
dsolve(diff(y(x),x$2)+2*(a*x+b)*diff(y(x),x)+(a^2*x^2+2*a*b*x+c)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {x \left (a x -2 \sqrt {b^{2}+a -c}+2 b \right )}{2}}+{\mathrm e}^{-\frac {x \left (a x +2 \sqrt {b^{2}+a -c}+2 b \right )}{2}} c_{2} \]
✓ Solution by Mathematica
Time used: 0.228 (sec). Leaf size: 86
DSolve[y''[x]+2*(a*x+b)*y'[x]+(a^2*x^2+2*a*b*x+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {1}{2} x \left (2 \sqrt {a+b^2-c}+a x+2 b\right )} \left (c_2 e^{2 x \sqrt {a+b^2-c}}+2 c_1 \sqrt {a+b^2-c}\right )}{2 \sqrt {a+b^2-c}} \]