Internal problem ID [8525]
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`]]
\[ \boxed {x^{m \left (n -1\right )+n} y^{\prime }-a y^{n}=b \,x^{n \left (m +1\right )}} \]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 62
dsolve(x^(m*(n-1)+n)*diff(y(x),x) - a*y(x)^n - b*x^(n*(m+1))=0,y(x), singsol=all)
\[ -x^{n \left (m +1\right )} \left (\int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{b \,x^{\left (1+n \right ) \left (m +1\right )}-\textit {\_a} \left (m +1\right ) x^{n \left (m +1\right )}+a \,x^{m +1} \textit {\_a}^{n}}d \textit {\_a} \right )+\ln \left (x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.386 (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]
\[ \text {Solve}\left [\int _1^{\left (\frac {a x^{-((m+1) n)}}{b}\right )^{\frac {1}{n}} y(x)}\frac {1}{K[1]^n-\left (\frac {b^{1-n} (m+1)^n}{a}\right )^{\frac {1}{n}} K[1]+1}dK[1]=b x^{m+1} \log (x) \left (\frac {a x^{-((m+1) n)}}{b}\right )^{\frac {1}{n}}+c_1,y(x)\right ] \]