28.1.81 problem 84
Internal
problem
ID
[4387]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
84
Date
solved
:
Monday, January 27, 2025 at 09:09:49 AM
CAS
classification
:
[_quadrature]
\begin{align*} x&=y^{\prime } \sqrt {1+{y^{\prime }}^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.067 (sec). Leaf size: 143
dsolve(x=diff(y(x),x)*sqrt( (diff(y(x),x))^2+1),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {i \left (-32 x^{4}-4 x^{2}+1\right ) \sinh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3 \sqrt {4 x^{2}+1}}-\frac {16 i x^{3} \cosh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3}+c_{1} \\
y \left (x \right ) &= \frac {i \left (-32 x^{4}-4 x^{2}+1\right ) \sinh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3 \sqrt {4 x^{2}+1}}+\frac {16 i x^{3} \cosh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3}+c_{1} \\
y \left (x \right ) &= -\frac {\left (\int \sqrt {2 \sqrt {4 x^{2}+1}-2}d x \right )}{2}+c_{1} \\
y \left (x \right ) &= \frac {\left (\int \sqrt {2 \sqrt {4 x^{2}+1}-2}d x \right )}{2}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.654 (sec). Leaf size: 207
DSolve[x==D[y[x],x]*Sqrt[ (D[y[x],x])^2+1],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {2} x \left (\sqrt {4 x^2+1}-2\right )}{3 \sqrt {\sqrt {4 x^2+1}-1}}+c_1 \\
y(x)\to \frac {\sqrt {2} x \left (\sqrt {4 x^2+1}-2\right )}{3 \sqrt {\sqrt {4 x^2+1}-1}}+c_1 \\
y(x)\to -\frac {\sqrt {2} x \left (4 x^2+3 \sqrt {4 x^2+1}+3\right )}{3 \left (-\sqrt {4 x^2+1}-1\right )^{3/2}}+c_1 \\
y(x)\to \frac {\sqrt {2} x \left (4 x^2+3 \sqrt {4 x^2+1}+3\right )}{3 \left (-\sqrt {4 x^2+1}-1\right )^{3/2}}+c_1 \\
\end{align*}