61.30.29 problem 177

Internal problem ID [12677]
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 : 177
Date solved : Tuesday, January 28, 2025 at 03:38:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 79

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

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

✓ Solution by Mathematica

Time used: 0.213 (sec). Leaf size: 93

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

\[ y(x)\to c_1 \cos \left (\frac {\sqrt {d} \log \left (-\sqrt {a} \sqrt {a x^2+2 b x+c}+a x+b\right )}{\sqrt {a}}\right )-c_2 \sin \left (\frac {\sqrt {d} \log \left (-\sqrt {a} \sqrt {a x^2+2 b x+c}+a x+b\right )}{\sqrt {a}}\right ) \]