60.1.395 problem 397

Internal problem ID [10409]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 397
Date solved : Monday, January 27, 2025 at 07:40:33 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.244 (sec). Leaf size: 132

dsolve(diff(y(x),x)^2-2*x^3*y(x)^2*diff(y(x),x)-4*x^2*y(x)^3 = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 2.828 (sec). Leaf size: 179

DSolve[-4*x^2*y[x]^3 - 2*x^3*y[x]^2*D[y[x],x] + D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {x^4 y(x)+4} y(x)^{3/2} \log \left (\sqrt {x^4 y(x)+4}-x^2 \sqrt {y(x)}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x y(x)^{3/2} \sqrt {x^4 y(x)+4} \log \left (\sqrt {x^4 y(x)+4}-x^2 \sqrt {y(x)}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to -\frac {4}{x^4} \\ \end{align*}