75.12.28 problem 302

Internal problem ID [16894]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 302
Date solved : Tuesday, January 28, 2025 at 09:40:01 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Solution by Maple

Time used: 1.355 (sec). Leaf size: 32

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

\[ \left (y^{n} x -n -2\right )^{n} y^{2 n} x^{-n}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.533 (sec). Leaf size: 160

DSolve[x^2*y[x]^n*D[y[x],x]==2*x*D[y[x],x]-y[x],y[x],x,IncludeSingularSolutions -> True]
 

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