61.27.33 problem 43

Internal problem ID [12543]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 43
Date solved : Tuesday, January 28, 2025 at 03:21:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.738 (sec). Leaf size: 41

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

\[ y = {\mathrm e}^{-a x} \left (c_{2} \left (\int \frac {{\mathrm e}^{-\frac {x^{3} \left (a x +\frac {4}{3}\right ) b}{4}}}{\left (a x +1\right )^{2}}d x \right )+c_{1} \right ) \left (a x +1\right ) \]

✓ Solution by Mathematica

Time used: 0.890 (sec). Leaf size: 84

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

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