60.1.188 problem 189

Internal problem ID [10202]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 189
Date solved : Monday, January 27, 2025 at 06:37:57 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.094 (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)
 

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

✓ Solution by Mathematica

Time used: 0.886 (sec). Leaf size: 91

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

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