60.1.186 problem 187

Internal problem ID [10200]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 187
Date solved : Monday, January 27, 2025 at 06:37:44 PM
CAS classification : [[_homogeneous, `class G`], _Riccati]

\begin{align*} x^{n} y^{\prime }-a y^{2}-b \,x^{2 n -2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 59

dsolve(x^n*diff(y(x),x) - a*y(x)^2 - b*x^(2*n-2)=0,y(x), singsol=all)
 

\[ y = \frac {x^{n -1} \left (n -1+\tan \left (\frac {\sqrt {4 a b -n^{2}+2 n -1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a b -n^{2}+2 n -1}\right )}{2 a} \]

✓ Solution by Mathematica

Time used: 0.521 (sec). Leaf size: 202

DSolve[x^n*D[y[x],x]- a*y[x]^2 - b*x^(2*n-2)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x^{n-1} \left (\left (-\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}+n-1\right ) x^{\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}}+c_1 \left (\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}+n-1\right )\right )}{2 a \left (x^{\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}}+c_1\right )} \\ y(x)\to \frac {x^{n-1} \left (\sqrt {a} \sqrt {b} \sqrt {\frac {(n-1)^2}{a b}-4}+n-1\right )}{2 a} \\ \end{align*}