60.1.52 problem 52

Internal problem ID [10066]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 52
Date solved : Monday, January 27, 2025 at 06:20:57 PM
CAS classification : [[_homogeneous, `class G`], _Chini]

\begin{align*} y^{\prime }-a y^{n}-b \,x^{\frac {n}{1-n}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.201 (sec). Leaf size: 65

dsolve(diff(y(x),x) - a*y(x)^n - b*x^(n/(1-n))=0,y(x), singsol=all)
 

\[ x^{\frac {n}{n -1}} \left (\int _{\textit {\_b}}^{y}\frac {1}{\textit {\_a}^{n} a \left (n -1\right ) x^{\frac {2 n -1}{n -1}}+x^{\frac {n}{n -1}} \textit {\_a} +b x \left (n -1\right )}d \textit {\_a} \right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 1.297 (sec). Leaf size: 117

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

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