61.32.19 problem 228

Internal problem ID [12729]
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 : 228
Date solved : Tuesday, January 28, 2025 at 04:16:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 156

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

\[ y = c_{1} \left (\frac {-i \sqrt {a b}+a x}{i \sqrt {a b}+a x}\right )^{\frac {i \sqrt {a b}\, c \sqrt {-a b}+a^{2} \sqrt {\frac {c^{2}-4 k}{a^{2}}}\, b}{4 a b \sqrt {-a b}}}+c_{2} \left (\frac {-i \sqrt {a b}+a x}{i \sqrt {a b}+a x}\right )^{\frac {i \sqrt {a b}\, c \sqrt {-a b}-a^{2} \sqrt {\frac {c^{2}-4 k}{a^{2}}}\, b}{4 a b \sqrt {-a b}}} \]

✓ Solution by Mathematica

Time used: 2.387 (sec). Leaf size: 178

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

\[ y(x)\to \exp \left (-\frac {1}{2} \int _1^x\frac {c+2 a K[2]}{a K[2]^2+b}dK[2]+\int _1^x\frac {2 a K[1]+i \sqrt {a} \sqrt {b} \sqrt {\frac {4 k-c^2}{a b}}}{2 a K[1]^2+2 b}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {2 a K[1]+i \sqrt {a} \sqrt {b} \sqrt {\frac {4 k-c^2}{a b}}}{2 a K[1]^2+2 b}dK[1]\right )dK[3]+c_1\right ) \]