60.1.85 problem 85

Internal problem ID [10099]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 85
Date solved : Tuesday, January 28, 2025 at 04:25:42 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }-x^{a -1} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.366 (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)
 

\[ y = {\left (\frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{-\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) \left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) \left (a^{\frac {1}{a}}\right )^{a} \textit {\_a} +a}d \textit {\_a} \right ) a^{2}+c_{1} a b -x^{a} b \right ) a -x^{a} b}{a}\right )}^{\frac {1}{b}} \]

✓ Solution by Mathematica

Time used: 0.330 (sec). Leaf size: 238

DSolve[D[y[x],x] - x^(a-1)*y[x]^(1-b)*f[x^a/a + y[x]^b/b]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]^{b-1}}{f\left (\frac {x^a}{a}+\frac {K[2]^b}{b}\right )+1}-\int _1^x\left (\frac {K[1]^{a-1} K[2]^{b-1} f''\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1}-\frac {f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right ) K[1]^{a-1} K[2]^{b-1} f''\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{\left (f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right ) K[1]^{a-1}}{f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right )+1}dK[1]=c_1,y(x)\right ] \]