60.1.328 problem 329

Internal problem ID [10342]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 329
Date solved : Monday, January 27, 2025 at 07:17:25 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Solution by Maple

Time used: 0.467 (sec). Leaf size: 72

dsolve(y(x)^m*x^n*(a*x*diff(y(x),x)+b*y(x))+alpha*x*diff(y(x),x)+beta*y(x) = 0,y(x), singsol=all)
 

\[ \left (x^{n} \left (a n -b m \right ) y^{m}-\beta m +\alpha n \right )^{-m \left (a \beta -b \alpha \right )} x^{\beta m \left (a n -b m \right )} \left (y^{m}\right )^{\alpha \left (a n -b m \right )}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.859 (sec). Leaf size: 119

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

\[ \text {Solve}\left [\frac {m \left (\beta (b m-a n) \log \left (n x^n (\alpha n-\beta m)\right )+n (a \beta -\alpha b) \log \left (x^n y(x)^m (b m-a n)+\beta m-\alpha n\right )\right )}{n (a n-b m) (\alpha n-\beta m)}-\frac {\alpha m \log (\alpha n y(x)-\beta m y(x))}{\alpha n-\beta m}=c_1,y(x)\right ] \]