29.14.5 problem 385

Internal problem ID [4984]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 385
Date solved : Monday, January 27, 2025 at 10:01:10 AM
CAS classification : [[_homogeneous, `class G`], _Riccati]

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

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 67

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

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

✓ Solution by Mathematica

Time used: 0.538 (sec). Leaf size: 162

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

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