61.30.14 problem 162

Internal problem ID [12662]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 162
Date solved : Tuesday, January 28, 2025 at 03:24:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 57

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

\[ y = c_{1} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )^{\frac {i \sqrt {c}}{\sqrt {a}}}+c_{2} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )^{-\frac {i \sqrt {c}}{\sqrt {a}}} \]

✓ Solution by Mathematica

Time used: 0.072 (sec). Leaf size: 62

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

\[ y(x)\to c_1 \cos \left (\frac {\sqrt {c} \text {arcsinh}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}}\right )+c_2 \sin \left (\frac {\sqrt {c} \text {arcsinh}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}}\right ) \]