Internal problem ID [7666]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 85.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-x^{a -1} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 152
dsolve(diff(y(x),x) - x^(a-1)*y(x)^(1-b)*f(x^a/a + y(x)^b/b)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (\frac {\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{-\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} \left (a^{\frac {1}{a}}\right )^{a} f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} \left (a^{\frac {1}{a}}\right )^{a} f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) \textit {\_a} +a}d \textit {\_a} \right ) a^{2}+c_{1} a b -x^{a} b \right ) a -x^{a} b}{a}\right )^{\frac {1}{b}} \]
✓ Solution by Mathematica
Time used: 0.309 (sec). Leaf size: 238
DSolve[y'[x] - x^(a-1)*y[x]^(1-b)*f[x^a/a + y[x]^b/b]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \operatorname {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]^{b-1}}{f\left (\frac {x^a}{a}+\frac {K[2]^b}{b}\right )+1}-\int _1^x\left (\frac {K[1]^{a-1} K[2]^{b-1} f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1}-\frac {f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right ) K[1]^{a-1} K[2]^{b-1} f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{\left (f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right ) K[1]^{a-1}}{f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right )+1}dK[1]=c_1,y(x)\right ] \]