60.1.214 problem 218
Internal
problem
ID
[10228]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
218
Date
solved
:
Sunday, March 30, 2025 at 03:32:51 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} \left (y-x^{2}\right ) y^{\prime }+4 x y&=0 \end{align*}
✓ Maple. Time used: 0.025 (sec). Leaf size: 57
ode:=(-x^2+y(x))*diff(y(x),x)+4*x*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {c_1 \sqrt {c_1^{2}-4 x^{2}}}{2}+\frac {c_1^{2}}{2}-x^{2} \\
y &= \frac {c_1 \sqrt {c_1^{2}-4 x^{2}}}{2}+\frac {c_1^{2}}{2}-x^{2} \\
\end{align*}
✓ Mathematica. Time used: 2.531 (sec). Leaf size: 246
ode=(y[x]-x^2)*D[y[x],x]+4*x*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to x^2 \left (1+\frac {2-2 i}{\frac {i \sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )-i}}-(1-i)}\right ) \\
y(x)\to x^2 \left (1+\frac {2-2 i}{(-1+i)-\frac {i \sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )-i}}}\right ) \\
y(x)\to x^2 \left (1+\frac {2-2 i}{(-1+i)-\frac {\sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )+i}}}\right ) \\
y(x)\to x^2 \left (1+\frac {2-2 i}{\frac {\sqrt {2}}{\sqrt {x^2 \cosh \left (\frac {2 c_1}{9}\right )+x^2 \sinh \left (\frac {2 c_1}{9}\right )+i}}-(1-i)}\right ) \\
y(x)\to 0 \\
y(x)\to -x^2 \\
\end{align*}
✓ Sympy. Time used: 0.946 (sec). Leaf size: 48
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x*y(x) + (-x**2 + y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = 2 C_{1}^{2} - 2 C_{1} \sqrt {\left (C_{1} - x\right ) \left (C_{1} + x\right )} - x^{2}, \ y{\left (x \right )} = 2 C_{1}^{2} + 2 C_{1} \sqrt {\left (C_{1} - x\right ) \left (C_{1} + x\right )} - x^{2}\right ]
\]