29.4.6 problem 94

Internal problem ID [4697]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 94
Date solved : Monday, January 27, 2025 at 09:32:09 AM
CAS classification : [[_homogeneous, `class G`], _Chini]

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

✓ Solution by Maple

Time used: 0.320 (sec). Leaf size: 65

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

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

✓ Solution by Mathematica

Time used: 1.302 (sec). Leaf size: 117

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

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