Internal problem ID [7769]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 189.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {x^{m \left (-1+n \right )+n} y^{\prime }-a y^{n}-b \,x^{n \left (m +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.085 (sec). Leaf size: 61
dsolve(x^(m*(n-1)+n)*diff(y(x),x) - a*y(x)^n - b*x^(n*(m+1))=0,y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \relax (x )}\frac {x^{n} x^{m n}}{-\left (b \,x^{m} x -\left (m +1\right ) \textit {\_a} \right ) x^{n} x^{m n}-a \,x^{m} x \,\textit {\_a}^{n}}d \textit {\_a} +\ln \relax (x )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.4 (sec). Leaf size: 91
DSolve[x^(m*(n-1)+n)*y'[x] - a*y[x]^n - b*x^(n*(m+1))==0,y[x],x,IncludeSingularSolutions -> True]
\[ \operatorname {Solve}\left [\int _1^{\left (\frac {a x^{-((m+1) n)}}{b}\right )^{\frac {1}{n}} y(x)}\frac {1}{K[1]^n-\left (\frac {b^{1-n} (m+1)^n}{a}\right )^{\frac {1}{n}} K[1]+1}dK[1]=b x^{m+1} \log (x) \left (\frac {a x^{-((m+1) n)}}{b}\right )^{\frac {1}{n}}+c_1,y(x)\right ] \]