61.32.28 problem 237

Internal problem ID [12738]
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 : 237
Date solved : Tuesday, January 28, 2025 at 04:17:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 270

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

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

✓ Solution by Mathematica

Time used: 2.743 (sec). Leaf size: 201

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

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