75.11.11 problem 270

Internal problem ID [16862]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 270
Date solved : Tuesday, January 28, 2025 at 09:35:37 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left (x y^{\prime }+y\right )^{2}&=y^{2} y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.143 (sec). Leaf size: 124

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

\begin{align*} y &= 4 x \\ y &= 0 \\ y &= -\frac {2 c_{1}^{2} \left (-c_{1} \sqrt {2}+x \right )}{-2 c_{1}^{2}+x^{2}} \\ y &= -\frac {2 c_{1}^{2} \left (c_{1} \sqrt {2}+x \right )}{-2 c_{1}^{2}+x^{2}} \\ y &= \frac {c_{1}^{3} \sqrt {2}-2 c_{1}^{2} x}{-2 c_{1}^{2}+4 x^{2}} \\ y &= \frac {c_{1}^{2} \left (c_{1} \sqrt {2}+2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.117 (sec). Leaf size: 61

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

\begin{align*} y(x)\to -\frac {4 e^{-2 c_1}}{-2+e^{2 c_1} x} \\ y(x)\to -\frac {4 e^{-2 c_1}}{2+e^{2 c_1} x} \\ y(x)\to 0 \\ y(x)\to 4 x \\ \end{align*}