61.31.16 problem 197

Internal problem ID [12697]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 197
Date solved : Tuesday, January 28, 2025 at 08:11:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.983 (sec). Leaf size: 2265

dsolve((a*x^3+b*x^2+c*x)*diff(y(x),x$2)+(-2*a*x^2-(b+1)*x+k)*diff(y(x),x)+2*(a*x+1)*y(x)=0,y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 5.846 (sec). Leaf size: 291

DSolve[(a*x^3+b*x^2+c*x)*D[y[x],{x,2}]+(-2*a*x^2-(b+1)*x+k)*D[y[x],x]+2*(a*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {\left (-k x (a x+2)-(b-1) x^2+c (k-2 x)+k^2\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {k^3-K[1] (b+3 a K[1]+3) k^2-K[1]^2 (a K[1]+1) (3 b+2 a K[1]-3) k-4 c^2 K[1]-(b-1) K[1]^3 (3 b+2 a K[1]-1)+c \left (k^2-K[1] (b+6 a K[1]+7) k-6 (b-1) K[1]^2\right )}{K[1] \left (-\left ((k-2 K[1]) c^2\right )-\left (k^2+(b-2) K[1] k+K[1]^2 (-3 b-2 a K[1]+1)\right ) c+K[1] \left (b^2 K[1]^2+a \left (-k^2+K[1] (a K[1]+2) k-K[1]^2\right ) K[1]+b \left (-k^2+K[1] (a K[1]+2) k+K[1]^2 (a K[1]-1)\right )\right )\right )}dK[1]\right )dK[2]+c_1\right )}{a k+b-c (k-2)-k^2+2 k-1} \]