60.1.172 problem 175
Internal
problem
ID
[10186]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
175
Date
solved
:
Sunday, March 30, 2025 at 03:24:32 PM
CAS
classification
:
[_linear]
\begin{align*} x \left (x^{2}-1\right ) y^{\prime }-\left (2 x^{2}-1\right ) y+a \,x^{3}&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 20
ode:=x*(x^2-1)*diff(y(x),x)-(2*x^2-1)*y(x)+a*x^3 = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x \left (\sqrt {x +1}\, c_1 \sqrt {x -1}+a \right )
\]
✓ Mathematica. Time used: 0.275 (sec). Leaf size: 93
ode=x*(x^2-1)*D[y[x],x] - (2*x^2-1)*y[x] + a*x^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \exp \left (\int _1^x\frac {1-2 K[1]^2}{K[1]-K[1]^3}dK[1]\right ) \left (\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1-2 K[1]^2}{K[1]-K[1]^3}dK[1]\right ) K[2]^2}{1-K[2]^2}dK[2]+c_1\right )
\]
✓ Sympy. Time used: 38.993 (sec). Leaf size: 338
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*x**3 + x*(x**2 - 1)*Derivative(y(x), x) - (2*x**2 - 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \begin {cases} \frac {C_{1} \sqrt {1 - x^{2}} \sqrt {x^{2} - 1}}{2 x \sqrt {1 - x^{2}} + 2 i x \sqrt {x^{2} - 1} - 2 \sqrt {1 - x^{2}} \sqrt {x^{2} - 1} \left (\begin {cases} - \frac {x}{\sqrt {x^{2} - 1}} & \text {for}\: \left |{x^{2}}\right | > 1 \\\frac {i x}{\sqrt {1 - x^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {1 - x^{2}} \sqrt {x^{2} - 1} \left (\begin {cases} - \frac {2 x}{\sqrt {x^{2} - 1}} + \frac {1}{x \sqrt {x^{2} - 1}} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {2 i x^{2} \sqrt {1 - x^{2}}}{x^{3} - x} + \frac {i \sqrt {1 - x^{2}}}{x^{3} - x} & \text {otherwise} \end {cases}\right )} + \frac {a \sqrt {1 - x^{2}}}{2 x \sqrt {1 - x^{2}} + 2 i x \sqrt {x^{2} - 1} - 2 \sqrt {1 - x^{2}} \sqrt {x^{2} - 1} \left (\begin {cases} - \frac {x}{\sqrt {x^{2} - 1}} & \text {for}\: \left |{x^{2}}\right | > 1 \\\frac {i x}{\sqrt {1 - x^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {1 - x^{2}} \sqrt {x^{2} - 1} \left (\begin {cases} - \frac {2 x}{\sqrt {x^{2} - 1}} + \frac {1}{x \sqrt {x^{2} - 1}} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {2 i x^{2} \sqrt {1 - x^{2}}}{x^{3} - x} + \frac {i \sqrt {1 - x^{2}}}{x^{3} - x} & \text {otherwise} \end {cases}\right )} & \text {for}\: x > -1 \wedge x < 1 \\\text {NaN} & \text {otherwise} \end {cases}
\]