61.28.16 problem 76

Internal problem ID [12576]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 76
Date solved : Tuesday, January 28, 2025 at 03:22:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+\left (b \,x^{3}+a^{2} x +a \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 39

dsolve(x*diff(y(x),x$2)-(2*a*x+1)*diff(y(x),x)+(b*x^3+a^2*x+a)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{a x +\frac {x^{2} \sqrt {-b}}{2}}+c_{2} {\mathrm e}^{a x -\frac {x^{2} \sqrt {-b}}{2}} \]

✓ Solution by Mathematica

Time used: 0.189 (sec). Leaf size: 59

DSolve[x*D[y[x],{x,2}]-(2*a*x+1)*D[y[x],x]+(b*x^3+a^2*x+a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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