61.30.26 problem 174

Internal problem ID [12674]
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 : 174
Date solved : Tuesday, January 28, 2025 at 03:37:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 43

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

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

✓ Solution by Mathematica

Time used: 0.155 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \cosh \left (m \log \left (\sqrt {2 a x+b+x^2}-a-x\right )\right )-i c_2 \sinh \left (m \log \left (\sqrt {2 a x+b+x^2}-a-x\right )\right ) \]