29.5.17 problem 133
Internal
problem
ID
[4734]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
5
Problem
number
:
133
Date
solved
:
Tuesday, January 28, 2025 at 02:39:40 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} y^{\prime }&=x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right ) \end{align*}
✓ Solution by Maple
Time used: 0.490 (sec). Leaf size: 173
dsolve(diff(y(x),x) = x^(m-1)*y(x)^(1-n)*f(a*x^m+b*y(x)^n),y(x), singsol=all)
\[
y \left (x \right ) = \left (\frac {-a \,x^{m}+\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} b n \textit {\_a} -\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} a m n +a \,m^{2}}d \textit {\_a} \right ) b \,m^{2}+c_{1} m -x^{m}\right ) b}{b}\right )^{\frac {1}{n}}
\]
✓ Solution by Mathematica
Time used: 0.543 (sec). Leaf size: 242
DSolve[D[y[x],x]==x^(m-1)*y[x]^(1-n)*f[a*x^m + b*y[x]^n],y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {a m K[2]^{n-1}}{a m+b n f\left (a x^m+b K[2]^n\right )}-\int _1^x\left (\frac {a b m n K[1]^{m-1} K[2]^{n-1} f''\left (a K[1]^m+b K[2]^n\right )}{a m+b n f\left (a K[1]^m+b K[2]^n\right )}-\frac {a b^2 m n^2 f\left (a K[1]^m+b K[2]^n\right ) K[1]^{m-1} K[2]^{n-1} f''\left (a K[1]^m+b K[2]^n\right )}{\left (a m+b n f\left (a K[1]^m+b K[2]^n\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {a m f\left (a K[1]^m+b y(x)^n\right ) K[1]^{m-1}}{a m+b n f\left (a K[1]^m+b y(x)^n\right )}dK[1]=c_1,y(x)\right ]
\]