77.1.82 problem 108 (page 162)
Internal
problem
ID
[17972]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
108
(page
162)
Date
solved
:
Tuesday, January 28, 2025 at 11:19:05 AM
CAS
classification
:
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
\begin{align*} a^{3} y^{\prime \prime \prime } y^{\prime \prime }&=\sqrt {1+c^{2} {y^{\prime \prime }}^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.402 (sec). Leaf size: 221
dsolve(a^3*diff(y(x),x$3)*diff(y(x),x$2)= sqrt(1+c^2* diff(y(x),x$2)^2),y(x), singsol=all)
\begin{align*}
y &= -\frac {i x^{2}}{2 c}+c_{1} x +c_{2} \\
y &= \frac {i x^{2}}{2 c}+c_{1} x +c_{2} \\
y &= \frac {-\int \left (a^{6} \ln \left (\sqrt {\left (x +c_{1} \right )^{2} c^{4}-a^{6}}+\left (x +c_{1} \right ) c^{2} \operatorname {csgn}\left (c^{2}\right )\right ) \operatorname {csgn}\left (c^{2}\right )-c^{2} \sqrt {\left (x +c_{1} \right )^{2} c^{4}-a^{6}}\, \left (x +c_{1} \right )\right )d x +\left (2 c_{2} x +2 c_{3} \right ) c^{3} a^{3}}{2 a^{3} c^{3}} \\
y &= \frac {\int \left (a^{6} \ln \left (\sqrt {\left (x +c_{1} \right )^{2} c^{4}-a^{6}}+\left (x +c_{1} \right ) c^{2} \operatorname {csgn}\left (c^{2}\right )\right ) \operatorname {csgn}\left (c^{2}\right )-c^{2} \sqrt {\left (x +c_{1} \right )^{2} c^{4}-a^{6}}\, \left (x +c_{1} \right )\right )d x +\left (2 c_{2} x +2 c_{3} \right ) c^{3} a^{3}}{2 a^{3} c^{3}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 25.445 (sec). Leaf size: 438
DSolve[a^3*D[y[x],{x,3}]*D[y[x],{x,2}]== Sqrt[1+c^2* D[y[x],{x,2}]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2} \left (a^6 \left (2+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2\right )-3 a^6 c^2 x \log \left (c^2 \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )\right )-3 a^9 c^2 c_1 \log \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )}{6 a^3 c^5}+c_3 x+c_2 \\
y(x)\to \frac {-\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2} \left (a^6 \left (2+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2\right )+3 a^6 c^2 x \log \left (c^2 \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )\right )+3 a^9 c^2 c_1 \log \left (c^2 \left (x+a^3 c_1\right )+\sqrt {a^6 \left (-1+c^4 c_1{}^2\right )+2 a^3 c^4 c_1 x+c^4 x^2}\right )}{6 a^3 c^5}+c_3 x+c_2 \\
\end{align*}