81.7.6 problem 7
Internal
problem
ID
[18703]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
VI.
Certain
particular
forms
of
equations.
Exercises
at
page
74
Problem
number
:
7
Date
solved
:
Tuesday, January 28, 2025 at 12:11:59 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}&=1 \end{align*}
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 55
dsolve(y(x)*diff(y(x),x$2)-(diff(y(x),x))^2=1,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}+{\mathrm e}^{\frac {-c_{2} -x}{c_{1}}}\right )}{2} \\
y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}+{\mathrm e}^{\frac {-c_{2} -x}{c_{1}}}\right )}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.931 (sec). Leaf size: 464
DSolve[y[x]*D[y[x],{x,2}]-(D[y[x],x])^2==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
\end{align*}