82.14.4 problem Ex. 4
Internal
problem
ID
[18731]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
III.
Equations
of
the
first
order
but
not
of
the
first
degree.
Problems
at
page
33
Problem
number
:
Ex.
4
Date
solved
:
Thursday, March 13, 2025 at 12:44:40 PM
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: 0.040 (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 \left (x \right ) &= \sqrt {a}\, x \\
y \left (x \right ) &= -\sqrt {a}\, x \\
y \left (x \right ) &= \frac {\left (\frac {x^{2}}{c_{1}^{2}}+a \right ) c_{1}}{2} \\
\end{align*}
✓ Mathematica. Time used: 18.175 (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: 1.973 (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)
\[
\left [ \log {\left (x \right )} = C_{1} + \begin {cases} - \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}, \ \log {\left (x \right )} = C_{1} + \begin {cases} \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}\right ]
\]