60.1.413 problem 415

Internal problem ID [10427]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 415
Date solved : Monday, January 27, 2025 at 07:43:14 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 89

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

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

✓ Solution by Mathematica

Time used: 0.525 (sec). Leaf size: 84

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

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