61.32.15 problem 224

Internal problem ID [12725]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 224
Date solved : Tuesday, January 28, 2025 at 04:16:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 47

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

\[ y = c_{1} \sin \left (\frac {\sqrt {c}\, \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )+c_{2} \cos \left (\frac {\sqrt {c}\, \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ) \]

✓ Solution by Mathematica

Time used: 3.154 (sec). Leaf size: 135

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

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