48.4.11 problem Problem 3.18
Internal
problem
ID
[7553]
Book
:
THEORY
OF
DIFFERENTIAL
EQUATIONS
IN
ENGINEERING
AND
MECHANICS.
K.T.
CHAU,
CRC
Press.
Boca
Raton,
FL.
2018
Section
:
Chapter
3.
Ordinary
Differential
Equations.
Section
3.6
Summary
and
Problems.
Page
218
Problem
number
:
Problem
3.18
Date
solved
:
Monday, January 27, 2025 at 03:05:43 PM
CAS
classification
:
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
\begin{align*} a y^{\prime \prime } y^{\prime \prime \prime }&=\sqrt {1+{y^{\prime \prime }}^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.082 (sec). Leaf size: 173
dsolve(a*diff(y(x),x$2)*diff(y(x),x$3)=sqrt(1+ diff(y(x),x$2)^2),y(x), singsol=all)
\begin{align*}
y &= -\frac {1}{2} i x^{2}+c_{1} x +c_{2} \\
y &= \frac {1}{2} i x^{2}+c_{1} x +c_{2} \\
y &= \frac {\left (2 a^{2}+\left (x +c_{1} \right )^{2}\right ) \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}-3 a \left (a \left (x +c_{1} \right ) \ln \left (x +c_{1} +\sqrt {\left (c_{1} +a +x \right ) \left (c_{1} -a +x \right )}\right )-2 c_{2} x -2 c_3 \right )}{6 a} \\
y &= \frac {\left (-2 a^{2}-\left (x +c_{1} \right )^{2}\right ) \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}+3 a \left (a \left (x +c_{1} \right ) \ln \left (x +c_{1} +\sqrt {\left (c_{1} +a +x \right ) \left (c_{1} -a +x \right )}\right )+2 c_{2} x +2 c_3 \right )}{6 a} \\
\end{align*}
✓ Solution by Mathematica
Time used: 11.588 (sec). Leaf size: 209
DSolve[a*D[y[x],{x,2}]*D[y[x],{x,3}]==Sqrt[1+ D[y[x],{x,2}]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2} \left (a^2 \left (2+c_1{}^2\right )+2 a c_1 x+x^2\right )}{6 a}-\frac {1}{2} a (x+a c_1) \log \left (\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}+a c_1+x\right )+c_3 x+c_2 \\
y(x)\to -\frac {\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2} \left (a^2 \left (2+c_1{}^2\right )+2 a c_1 x+x^2\right )}{6 a}+\frac {1}{2} a (x+a c_1) \log \left (\sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}+a c_1+x\right )+c_3 x+c_2 \\
\end{align*}