75.8.17 problem 215
Internal
problem
ID
[16833]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
8.
First
order
not
solved
for
the
derivative.
Exercises
page
67
Problem
number
:
215
Date
solved
:
Tuesday, January 28, 2025 at 09:34:19 AM
CAS
classification
:
[_quadrature]
\begin{align*} x \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}}&=a \end{align*}
✓ Solution by Maple
Time used: 0.106 (sec). Leaf size: 229
dsolve(x*(1+diff(y(x),x)^2)^(3/2)=a,y(x), singsol=all)
\begin{align*}
y &= \int \frac {\sqrt {\left (a \,x^{2}\right )^{{2}/{3}}-x^{2}}}{x}d x +c_{1} \\
y &= -\frac {\left (\int \frac {\sqrt {-2 i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-2 \left (a \,x^{2}\right )^{{2}/{3}}-4 x^{2}}}{x}d x \right )}{2}+c_{1} \\
y &= \frac {\left (\int \frac {\sqrt {-2 i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-2 \left (a \,x^{2}\right )^{{2}/{3}}-4 x^{2}}}{x}d x \right )}{2}+c_{1} \\
y &= -\int \frac {\sqrt {\left (a \,x^{2}\right )^{{2}/{3}}-x^{2}}}{x}d x +c_{1} \\
y &= -\frac {\sqrt {2}\, \left (\int \frac {\sqrt {i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-\left (a \,x^{2}\right )^{{2}/{3}}-2 x^{2}}}{x}d x \right )}{2}+c_{1} \\
y &= \frac {\sqrt {2}\, \left (\int \frac {\sqrt {i \sqrt {3}\, \left (a \,x^{2}\right )^{{2}/{3}}-\left (a \,x^{2}\right )^{{2}/{3}}-2 x^{2}}}{x}d x \right )}{2}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.343 (sec). Leaf size: 216
DSolve[x*(1+D[y[x],x]^2)^(3/2)==a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1 \\
y(x)\to x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1 \\
y(x)\to c_1-x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2} \\
y(x)\to x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1 \\
y(x)\to c_1-x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2} \\
y(x)\to x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1 \\
\end{align*}