Internal problem ID [2882]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 5
Problem number: 133.
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^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.13 (sec). Leaf size: 174
dsolve(diff(y(x),x) = x^(m-1)*y(x)^(1-n)*f(a*x^m+b*y(x)^n),y(x), singsol=all)
\[ y \relax (x ) = \left (-\frac {-\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} b n \textit {\_a} -\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} a m n +a \,m^{2}}d \textit {\_a} \right ) b \,m^{2}+c_{1} m -x^{m}\right ) b +a \,x^{m}}{b}\right )^{\frac {1}{n}} \]
✓ Solution by Mathematica
Time used: 0.406 (sec). Leaf size: 242
DSolve[y'[x]==x^(m-1) y[x]^(1-n) f[a x^m + b y[x]^n],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {a m K[2]^{n-1}}{a m+b n f\left (a x^m+b K[2]^n\right )}-\int _1^x\left (\frac {a b m n K[1]^{m-1} K[2]^{n-1} f'\left (a K[1]^m+b K[2]^n\right )}{a m+b n f\left (a K[1]^m+b K[2]^n\right )}-\frac {a b^2 m n^2 f\left (a K[1]^m+b K[2]^n\right ) K[1]^{m-1} K[2]^{n-1} f'\left (a K[1]^m+b K[2]^n\right )}{\left (a m+b n f\left (a K[1]^m+b K[2]^n\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {a m f\left (a K[1]^m+b y(x)^n\right ) K[1]^{m-1}}{a m+b n f\left (a K[1]^m+b y(x)^n\right )}dK[1]=c_1,y(x)\right ] \]