Internal problem ID [10870]
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: 36.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+a \,x^{2}+b +2 x \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 57
dsolve(diff(y(x),x$2)+(2*x^2+a)*diff(y(x),x)+(x^4+a*x^2+2*x+b)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {x \left (2 x^{2}-3 \sqrt {a^{2}-4 b}+3 a \right )}{6}}+c_{2} {\mathrm e}^{-\frac {x \left (2 x^{2}+3 \sqrt {a^{2}-4 b}+3 a \right )}{6}} \]
✓ Solution by Mathematica
Time used: 0.217 (sec). Leaf size: 79
DSolve[y''[x]+(2*x^2+a)*y'[x]+(x^4+a*x^2+2*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {1}{6} x \left (3 \sqrt {a^2-4 b}+3 a+2 x^2\right )} \left (c_2 e^{x \sqrt {a^2-4 b}}+c_1 \sqrt {a^2-4 b}\right )}{\sqrt {a^2-4 b}} \]