60.7.39 problem 1629 (6.39)

Internal problem ID [11628]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1629 (6.39)
Date solved : Tuesday, January 28, 2025 at 06:07:13 PM
CAS classification : [[_2nd_order, _with_potential_symmetries]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 38

dsolve(diff(diff(y(x),x),x)+(3*y(x)+f(x))*diff(y(x),x)+y(x)^3+y(x)^2*f(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 54.964 (sec). Leaf size: 82

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

\begin{align*} y(x)\to \frac {\int _1^x\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) c_1dK[2]+c_2}{\int _1^x\int _1^{K[5]}\exp \left (-\int _1^{K[4]}f(K[3])dK[3]\right ) c_1dK[4]dK[5]+c_2 x+1} \\ y(x)\to \frac {1}{x} \\ \end{align*}