60.7.51 problem 1641 (6.51)

Internal problem ID [11640]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1641 (6.51)
Date solved : Monday, January 27, 2025 at 11:28:55 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 29

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

\[ \int _{}^{y}{\mathrm e}^{\int f \left (\textit {\_b} \right )d \textit {\_b}}d \textit {\_b} -c_{1} \left (\int {\mathrm e}^{-\int g \left (x \right )d x}d x \right )-c_{2} = 0 \]

✓ Solution by Mathematica

Time used: 1.228 (sec). Leaf size: 61

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

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[3]}-f(K[1])dK[1]\right )dK[3]\&\right ]\left [\int _1^x-\exp \left (-\int _1^{K[4]}g(K[2])dK[2]\right ) c_1dK[4]+c_2\right ] \]