60.7.168 problem 1759 (book 6.168)

Internal problem ID [11757]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1759 (book 6.168)
Date solved : Monday, January 27, 2025 at 11:34:14 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 73

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

\begin{align*} y &= -\frac {b}{a} \\ y &= \frac {\left (\frac {1}{\left (a +c \right ) \left (c_{1} x +c_{2} \right )}\right )^{\frac {c}{a +c}} \left (-\left (\frac {1}{\left (a +c \right ) \left (c_{1} x +c_{2} \right )}\right )^{-\frac {c}{a +c}} b +\left (a +c \right ) \left (c_{1} x +c_{2} \right )\right )}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 8.823 (sec). Leaf size: 31

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

\[ y(x)\to \frac {-b+(c_1 (a+c) (x+c_2)){}^{\frac {a}{a+c}}}{a} \]