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
:
Sunday, March 30, 2025 at 03:12:00 AM
CAS
classification
:
[_quadrature]
\begin{align*} x&=y^{\prime } \sqrt {{y^{\prime }}^{2}+1} \end{align*}
✓ Maple. Time used: 0.064 (sec). Leaf size: 143
ode:=x = diff(y(x),x)*(1+diff(y(x),x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\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 &= \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 &= -\frac {\int \sqrt {-2+2 \sqrt {4 x^{2}+1}}d x}{2}+c_1 \\
y &= \frac {\int \sqrt {-2+2 \sqrt {4 x^{2}+1}}d x}{2}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 1.654 (sec). Leaf size: 207
ode=x==D[y[x],x]*Sqrt[ (D[y[x],x])^2+1];
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 4.456 (sec). Leaf size: 966
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x - sqrt(Derivative(y(x), x)**2 + 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\text {Solution too large to show}
\]