29.30.8 problem 867
Internal
problem
ID
[5448]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
30
Problem
number
:
867
Date
solved
:
Sunday, March 30, 2025 at 08:13:46 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}-2 y y^{\prime }+a x&=0 \end{align*}
✓ Maple. Time used: 0.094 (sec). Leaf size: 33
ode:=x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)+a*x = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \sqrt {a}\, x \\
y &= -\sqrt {a}\, x \\
y &= \frac {\left (\frac {x^{2}}{c_1^{2}}+a \right ) c_1}{2} \\
\end{align*}
✓ Mathematica. Time used: 19.313 (sec). Leaf size: 519
ode=x (D[y[x],x])^2-2 y[x] D[y[x],x]+a x==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {a} x \tan (c_1-i \log (x))}{\sqrt {\sec ^2(c_1-i \log (x))}} \\
y(x)\to \frac {\sqrt {a} x \tan (c_1-i \log (x))}{\sqrt {\sec ^2(c_1-i \log (x))}} \\
y(x)\to -\frac {\sqrt {a} x \tan (i \log (x)+c_1)}{\sqrt {\sec ^2(i \log (x)+c_1)}} \\
y(x)\to \frac {\sqrt {a} x \tan (i \log (x)+c_1)}{\sqrt {\sec ^2(i \log (x)+c_1)}} \\
y(x)\to -\sqrt {a} x \\
y(x)\to \sqrt {a} x \\
y(x)\to \frac {i \sqrt {a} \left (e^{2 i \text {Interval}[\{0,2 \pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}-x^4 e^{2 i \text {Interval}[\{0,\pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}\right )}{2 x} \\
y(x)\to \frac {i \sqrt {a} \left (x^4 e^{2 i \text {Interval}[\{0,\pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}-e^{2 i \text {Interval}[\{0,2 \pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}\right )}{2 x} \\
y(x)\to \frac {i \sqrt {a} \left (x^4 e^{2 i \text {Interval}[\{0,\pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}-e^{2 i \text {Interval}[\{0,2 \pi \}]} \sqrt {\frac {x^2 e^{2 i \text {Interval}[\{0,\pi \}]}}{\left (x^2+e^{2 i \text {Interval}[\{0,\pi \}]}\right )^2}}\right )}{2 x} \\
\end{align*}
✓ Sympy. Time used: 2.127 (sec). Leaf size: 82
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*x + x*Derivative(y(x), x)**2 - 2*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} + \begin {cases} - \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}, \ \log {\left (x \right )} = C_{1} + \begin {cases} \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}\right ]
\]