61.28.27 problem 87

Internal problem ID [12587]
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 : 87
Date solved : Tuesday, January 28, 2025 at 03:22:22 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.543 (sec). Leaf size: 42

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

\[ y = x \,{\mathrm e}^{-\frac {x \left (a \,x^{2}+3 b \right )}{3}} \left (c_{1} \left (\int \frac {{\mathrm e}^{\frac {x \left (a \,x^{2}+3 b \right )}{3}}}{x^{2}}d x \right )+c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.511 (sec). Leaf size: 93

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

\[ y(x)\to \exp \left (\int _1^x\left (-\frac {1}{2} a K[1]^2-\frac {b}{2}+\frac {1}{K[1]}\right )dK[1]-\frac {1}{6} x \left (a x^2+3 b\right )\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\left (-\frac {1}{2} a K[1]^2-\frac {b}{2}+\frac {1}{K[1]}\right )dK[1]\right )dK[2]+c_1\right ) \]