Internal
problem
ID
[7774] Book
:
Own
collection
of
miscellaneous
problems Section
:
section
1.0 Problem
number
:
81 Date
solved
:
Monday, October 21, 2024 at 04:18:25 PM CAS
classification
:
[_quadrature]
The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(x=0\) is
\[
\{0<y\}
\]
But the point \(y_0 = 0\) is not inside this domain. Hence existence and
uniqueness theorem does not apply. Solution exists but no guarantee that unique solution
exists.
1.82.2 Solved as first order quadrature ode
Time used: 0.043 (sec)
Since the ode has the form \(y^{\prime }=f(y)\) and initial conditions \(\left (x_0,y_0\right ) \) are given such that they satisfy the ode
itself, then we can write
for \(y\). This is because we had to divide by this in the above step. This gives the following
singular solution(s), which also have to satisfy the given ODE.
We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that
satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives
But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that
If the above condition is satisfied,
then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know
now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not
satisfied then this method will not work and we have to now look for an integrating
factor to force this condition, which might or might not exist. The first step is
to write the ODE in standard form to check for exactness, which is
Since \(\frac {\partial M}{\partial y} \neq \frac {\partial N}{\partial x}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an
integrating factor to make it exact. Let
\(M\) and \(N\) are now multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\)
and \(\overline {N}\) so not to confuse them with the original \(M\) and \(N\).
Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function
of both \(x\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives
Where \(c_1\) is constant of integration. Substituting result found above for \(f(y)\) into
equation (3) gives \(\phi \)
\[
\phi = -2 x +2 \sqrt {y}+ c_1
\]
But since \(\phi \) itself is a constant function, then let \(\phi =c_2\) where \(c_2\) is new
constant and combining \(c_1\) and \(c_2\) constants into the constant \(c_1\) gives the solution as
\[
c_1 = -2 x +2 \sqrt {y}
\]
Solving for the constant of integration from initial conditions, the solution becomes