60.1.549 problem 552

Internal problem ID [10563]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 552
Date solved : Monday, January 27, 2025 at 09:05:39 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 43

dsolve(diff(y(x),x)^n-f(x)*g(y(x))=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.279 (sec). Leaf size: 41

DSolve[-(f[x]*g[y[x]]) + D[y[x],x]^n==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}g(K[1])^{-1/n}dK[1]\&\right ]\left [\int _1^xf(K[2])^{\frac {1}{n}}dK[2]+c_1\right ] \]