60.1.126 problem 127

Internal problem ID [10140]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 127
Date solved : Monday, January 27, 2025 at 06:29:43 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.114 (sec). Leaf size: 39

dsolve(x*diff(y(x),x) - y(x)*f(x^a*y(x)^b)=0,y(x), singsol=all)
 

\[ \int _{\textit {\_b}}^{y}\frac {1}{\left (f \left (x^{a} \textit {\_a}^{b}\right ) b +a \right ) \textit {\_a}}d \textit {\_a} -\frac {\ln \left (x \right )}{b}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.315 (sec). Leaf size: 186

DSolve[x*D[y[x],x] - y[x]*f[x^a*y[x]^b]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {b}{\left (a+b f\left (x^a K[2]^b\right )\right ) K[2]}-\int _1^x\left (\frac {b^2 K[1]^{a-1} K[2]^{b-1} f''\left (K[1]^a K[2]^b\right )}{a+b f\left (K[1]^a K[2]^b\right )}-\frac {b^3 f\left (K[1]^a K[2]^b\right ) K[1]^{a-1} K[2]^{b-1} f''\left (K[1]^a K[2]^b\right )}{\left (a+b f\left (K[1]^a K[2]^b\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {b f\left (K[1]^a y(x)^b\right )}{\left (a+b f\left (K[1]^a y(x)^b\right )\right ) K[1]}dK[1]=c_1,y(x)\right ] \]