82.23.1 problem Ex. 1
Internal
problem
ID
[18857]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
IV.
Singular
solutions.
problems
on
chapter
IV.
page
49
Problem
number
:
Ex.
1
Date
solved
:
Tuesday, January 28, 2025 at 12:30:58 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}-2 y^{\prime } y+a x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.040 (sec). Leaf size: 33
dsolve(x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)+a*x=0,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*}
✓ Solution by Mathematica
Time used: 12.567 (sec). Leaf size: 519
DSolve[x*D[y[x],x]^2-2*y[x]*D[y[x],x]+a*x==0,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*}