Internal
problem
ID
[18248] Book
:
Elementary
Differential
Equations.
By
Thornton
C.
Fry.
D
Van
Nostrand.
NY.
First
Edition
(1929) Section
:
Chapter
IV.
Methods
of
solution:
First
order
equations.
section
33.
Problems
at
page
91 Problem
number
:
9
(b) Date
solved
:
Monday, December 23, 2024 at 09:19:14 PM CAS
classification
:
[_separable]
Solve
\begin{align*} \sqrt {-u^{2}+1}\, v^{\prime }&=2 u \sqrt {1-v^{2}} \end{align*}
Solved as first order separable ode
Time used: 0.352 (sec)
The ode \(v^{\prime } = \frac {2 u \sqrt {1-v^{2}}}{\sqrt {-u^{2}+1}}\) is separable as it can be written as
We now need to find the singular solutions, these are found by finding for what values \(g(v)\) is
zero, since we had to divide by this above. Solving \(g(v)=0\) or \(\sqrt {-v^{2}+1}=0\) for \(v\) gives
\begin{align*} v&=-1\\ v&=1 \end{align*}
Now we go over each such singular solution and check if it verifies the ode itself and
any initial conditions given. If it does not then the singular solution will not be
used.
Therefore the solutions found are
\begin{align*} \arcsin \left (v\right ) = -2 \sqrt {-u^{2}+1}+c_{1}\\ v = -1\\ v = 1 \end{align*}
Solving for \(v\) gives
\begin{align*}
v &= -1 \\
v &= 1 \\
v &= \sin \left (-2 \sqrt {-u^{2}+1}+c_{1} \right ) \\
\end{align*}
Summary of solutions found
\begin{align*}
v &= -1 \\
v &= 1 \\
v &= \sin \left (-2 \sqrt {-u^{2}+1}+c_{1} \right ) \\
\end{align*}
`Methodsfor first order ODEs:---Trying classification methods ---tryinga quadraturetrying1st order lineartryingBernoullitryingseparable<-separable successful`