60.1.467 problem 470

Internal problem ID [10481]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 470
Date solved : Monday, January 27, 2025 at 07:55:00 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.215 (sec). Leaf size: 87

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

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

✓ Solution by Mathematica

Time used: 1.108 (sec). Leaf size: 178

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

\begin{align*} \text {Solve}\left [\frac {1}{2} \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{x \sqrt {x^4+4 y(x)^2}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{x \sqrt {x^4+4 y(x)^2}}+\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to -\frac {i x^2}{2} \\ y(x)\to \frac {i x^2}{2} \\ \end{align*}