29.29.2 problem 824

Internal problem ID [5407]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 824
Date solved : Monday, January 27, 2025 at 11:21:09 AM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} {y^{\prime }}^{2}+x y^{2} y^{\prime }+y^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.294 (sec). Leaf size: 124

dsolve(diff(y(x),x)^2+x*y(x)^2*diff(y(x),x)+y(x)^3 = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.959 (sec). Leaf size: 152

DSolve[(D[y[x],x])^2+x*y[x]^2* D[y[x],x]+y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [-\frac {\sqrt {y(x)} \sqrt {4-x^2 y(x)} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)}\right )}{\sqrt {-y(x)} \sqrt {x^2 y(x)-4}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {y(x)} \sqrt {4-x^2 y(x)} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)}\right )}{\sqrt {-y(x)} \sqrt {x^2 y(x)-4}}-\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to \frac {4}{x^2} \\ \end{align*}