60.1.512 problem 525
Internal
problem
ID
[10526]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
525
Date
solved
:
Sunday, March 30, 2025 at 05:44:59 PM
CAS
classification
:
[_separable]
\begin{align*} {y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.047 (sec). Leaf size: 98
ode:=diff(y(x),x)^2-a*x*y(x)*diff(y(x),x)+2*a*y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= c_1 {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{-2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{\frac {x \left (a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\
y &= c_1 {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{-\frac {x \left (-a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\
\end{align*}
✓ Mathematica. Time used: 0.496 (sec). Leaf size: 195
ode=2*a*y[x]^2 - a*x*y[x]*D[y[x],x] + D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to c_1 \exp \left (\frac {-8 \sqrt {a x^2-8} \arcsin \left (\frac {\sqrt {a} x}{2 \sqrt {2}}\right )+a x^2 \sqrt {8-a x^2}-\sqrt {a} x \sqrt {-\left (a x^2-8\right )^2}}{4 \sqrt {8-a x^2}}\right ) \\
y(x)\to c_1 \exp \left (\frac {8 \sqrt {a x^2-8} \arcsin \left (\frac {\sqrt {a} x}{2 \sqrt {2}}\right )+a x^2 \sqrt {8-a x^2}+\sqrt {a} x \sqrt {-\left (a x^2-8\right )^2}}{4 \sqrt {8-a x^2}}\right ) \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 5.049 (sec). Leaf size: 289
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*x*y(x)*Derivative(y(x), x) + 2*a*y(x)**2 + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} e^{C_{1} - \frac {a^{\frac {3}{2}} x^{3}}{4 \sqrt {a x^{2} - 8}} + \frac {2 \sqrt {a} x}{\sqrt {a x^{2} - 8}} + \frac {a x^{2}}{4} + 2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {a} x}{4} \right )}} & \text {for}\: \left |{a x^{2}}\right | > 8 \\e^{C_{1} + \frac {i a^{\frac {3}{2}} x^{3}}{4 \sqrt {- a x^{2} + 8}} - \frac {2 i \sqrt {a} x}{\sqrt {- a x^{2} + 8}} + \frac {a x^{2}}{4} - 2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {a} x}{4} \right )}} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} e^{C_{1} + \frac {a^{\frac {3}{2}} x^{3}}{4 \sqrt {a x^{2} - 8}} - \frac {2 \sqrt {a} x}{\sqrt {a x^{2} - 8}} + \frac {a x^{2}}{4} - 2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {a} x}{4} \right )}} & \text {for}\: \left |{a x^{2}}\right | > 8 \\e^{C_{1} - \frac {i a^{\frac {3}{2}} x^{3}}{4 \sqrt {- a x^{2} + 8}} + \frac {2 i \sqrt {a} x}{\sqrt {- a x^{2} + 8}} + \frac {a x^{2}}{4} + 2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {a} x}{4} \right )}} & \text {otherwise} \end {cases}\right ]
\]