60.7.80 problem 1671 (book 6.80)

Internal problem ID [11669]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1671 (book 6.80)
Date solved : Tuesday, January 28, 2025 at 06:07:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.105 (sec). Leaf size: 38

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

\[ y = \frac {\left (i \sqrt {b}\, \left (\int \frac {\tan \left (-i \sqrt {a}\, \sqrt {b}\, x +c_{1} \right )}{x^{2}}d x \right )+c_{2} \sqrt {a}\right ) x}{\sqrt {a}} \]

✓ Solution by Mathematica

Time used: 120.332 (sec). Leaf size: 50

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

\[ y(x)\to x \left (\int _1^x\frac {\sqrt {-\frac {b}{a}} \tan \left (c_1+\frac {b K[2]}{\sqrt {-\frac {b}{a}}}\right )}{K[2]^2}dK[2]+c_2\right ) \]