Internal problem ID [7892]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 312.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (x +y y^{\prime }\right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 236
dsolve((y(x)^2/b+x^2/a)*(y(x)*diff(y(x),x)+x)+(a-b)/(a+b)*(y(x)*diff(y(x),x)-x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {a \left (-b \,x^{2}+a b +{\mathrm e}^{-\frac {2 \LambertW \left (\frac {\left (a +b \right ) {\mathrm e}^{-\frac {x^{2}}{2 b}} {\mathrm e}^{\frac {b \,x^{2}}{2 a^{2}}} {\mathrm e}^{-\frac {1}{2}} {\mathrm e}^{-\frac {b}{2 a}} {\mathrm e}^{-\frac {c_{1}}{a b}}}{2 a^{2} b}\right ) a^{2} b +a^{2} x^{2}-b^{2} x^{2}+a^{2} b +a \,b^{2}+2 c_{1} a}{2 a^{2} b}}\right )}}{a} \\ y \relax (x ) = -\frac {\sqrt {a \left (-b \,x^{2}+a b +{\mathrm e}^{-\frac {2 \LambertW \left (\frac {\left (a +b \right ) {\mathrm e}^{-\frac {x^{2}}{2 b}} {\mathrm e}^{\frac {b \,x^{2}}{2 a^{2}}} {\mathrm e}^{-\frac {1}{2}} {\mathrm e}^{-\frac {b}{2 a}} {\mathrm e}^{-\frac {c_{1}}{a b}}}{2 a^{2} b}\right ) a^{2} b +a^{2} x^{2}-b^{2} x^{2}+a^{2} b +a \,b^{2}+2 c_{1} a}{2 a^{2} b}}\right )}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 61.302 (sec). Leaf size: 176
DSolve[((a - b)*(-x + y[x]*y'[x]))/(a + b) + (x^2/a + y[x]^2/b)*(x + y[x]*y'[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {b} \sqrt {(a+b) \left (a-x^2\right )+2 a^2 \text {ProductLog}\left (\frac {c_1 (a+b) \exp \left (-\frac {(a+b) \left (x^2 (a-b)+a b\right )}{2 a^2 b}\right )}{2 a^3 b^2}\right )}}{\sqrt {a} \sqrt {a+b}} \\ y(x)\to \frac {\sqrt {b} \sqrt {(a+b) \left (a-x^2\right )+2 a^2 \text {ProductLog}\left (\frac {c_1 (a+b) \exp \left (-\frac {(a+b) \left (x^2 (a-b)+a b\right )}{2 a^2 b}\right )}{2 a^3 b^2}\right )}}{\sqrt {a} \sqrt {a+b}} \\ \end{align*}