61.27.26 problem 36

Internal problem ID [12536]
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
Problem number : 36
Date solved : Tuesday, January 28, 2025 at 03:20:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+a \,x^{2}+b +2 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (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: 0.149 (sec). Leaf size: 79

DSolve[D[y[x],{x,2}]+(2*x^2+a)*D[y[x],x]+(x^4+a*x^2+2*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-\frac {1}{6} x \left (3 \sqrt {a^2-4 b}+3 a+2 x^2\right )} \left (c_2 e^{x \sqrt {a^2-4 b}}+c_1 \sqrt {a^2-4 b}\right )}{\sqrt {a^2-4 b}} \]