Internal
problem
ID
[8495] Book
:
Second
order
enumerated
odes Section
:
section
1 Problem
number
:
12 Date
solved
:
Sunday, November 10, 2024 at 03:55:05 AM CAS
classification
:
[[_2nd_order, _missing_x]]
Substituting the above values for \(p,q,r,s\) gives
\begin{align*} y^{\prime }+y&=c_1 \end{align*}
We now have a first order ode to solve which is
\begin{align*} y^{\prime }+y = c_1 \end{align*}
Integrating gives
\begin{align*} \int \frac {1}{-y +c_1}d y &= dx\\ -\ln \left (-y +c_1 \right )&= x +c_2 \end{align*}
Singular solutions are found by solving
\begin{align*} -y +c_1&= 0 \end{align*}
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.
\begin{align*} y = c_1 \end{align*}
Solving for \(y\) gives
\begin{align*}
y &= -{\mathrm e}^{-x -c_2}+c_1 \\
\end{align*}
Will add steps showing solving for IC soon.
Summary of solutions found
\begin{align*}
y &= c_1 \\
y &= -{\mathrm e}^{-x -c_2}+c_1 \\
\end{align*}
Solved as second order missing y ode
Time used: 0.099 (sec)
This is second order ode with missing dependent variable \(y\). Let
Which is now solve for \(p(x)\) as first order ode. Integrating gives
\begin{align*} \int -\frac {1}{p}d p &= dx\\ -\ln \left (p \right )&= x +c_1 \end{align*}
Singular solutions are found by solving
\begin{align*} -p&= 0 \end{align*}
for \(p \left (x \right )\). 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.
\begin{align*} p \left (x \right ) = 0 \end{align*}
\begin{align*} \int \frac {1}{-y +c_1}d y &= dx\\ -\ln \left (-y +c_1 \right )&= x +c_2 \end{align*}
Singular solutions are found by solving
\begin{align*} -y +c_1&= 0 \end{align*}
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.
\begin{align*} y = c_1 \end{align*}
Solving for \(y\) gives
\begin{align*}
y &= -{\mathrm e}^{-x -c_2}+c_1 \\
\end{align*}
Will add steps showing solving for IC soon.
Summary of solutions found
\begin{align*}
y &= c_1 \\
y &= -{\mathrm e}^{-x -c_2}+c_1 \\
\end{align*}
Solved as second order integrable as is ode (ABC method)
Time used: 0.049 (sec)
Writing the ode as
\[
y^{\prime \prime }+y^{\prime } = 0
\]
Integrating both sides of the ODE w.r.t \(x\) gives
\begin{align*} \int \frac {1}{-y +c_1}d y &= dx\\ -\ln \left (-y +c_1 \right )&= x +c_2 \end{align*}
Singular solutions are found by solving
\begin{align*} -y +c_1&= 0 \end{align*}
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.
\begin{align*} y = c_1 \end{align*}
Solving for \(y\) gives
\begin{align*}
y &= -{\mathrm e}^{-x -c_2}+c_1 \\
\end{align*}
Will add steps showing solving for IC soon.
Solved as second order ode using Kovacic algorithm
Time used: 0.040 (sec)
Writing the ode as
\begin{align*} y^{\prime \prime }+y^{\prime } &= 0 \tag {1} \\ A y^{\prime \prime } + B y^{\prime } + C y &= 0 \tag {2} \end{align*}
Comparing (1) and (2) shows that
\begin{align*} A &= 1 \\ B &= 1\tag {3} \\ C &= 0 \end{align*}
Applying the Liouville transformation on the dependent variable gives
\begin{align*} z(x) &= y e^{\int \frac {B}{2 A} \,dx} \end{align*}
Then (2) becomes
\begin{align*} z''(x) = r z(x)\tag {4} \end{align*}
Where \(r\) is given by
\begin{align*} r &= \frac {s}{t}\tag {5} \\ &= \frac {2 A B' - 2 B A' + B^2 - 4 A C}{4 A^2} \end{align*}
Substituting the values of \(A,B,C\) from (3) in the above and simplifying gives
\begin{align*} r &= \frac {1}{4}\tag {6} \end{align*}
Equation (7) is now solved. After finding \(z(x)\) then \(y\) is found using the inverse transformation
\begin{align*} y &= z \left (x \right ) e^{-\int \frac {B}{2 A} \,dx} \end{align*}
The first step is to determine the case of Kovacic algorithm this ode belongs to. There are 3
cases depending on the order of poles of \(r\) and the order of \(r\) at \(\infty \). The following table
summarizes these cases.
Need to have at least one pole
that is either order \(2\) or odd order
greater than \(2\). Any other pole order
is allowed as long as the above
condition is satisfied. Hence the
following set of pole orders are all
allowed. \(\{1,2\}\),\(\{1,3\}\),\(\{2\}\),\(\{3\}\),\(\{3,4\}\),\(\{1,2,5\}\).
no condition
3
\(\left \{ 1,2\right \} \)
\(\left \{ 2,3,4,5,6,7,\cdots \right \} \)
Table 2.5: Necessary conditions for each Kovacic case
The order of \(r\) at \(\infty \) is the degree of \(t\) minus the degree of \(s\). Therefore
There are no poles in \(r\). Therefore the set of poles \(\Gamma \) is empty. Since there is no odd order pole
larger than \(2\) and the order at \(\infty \) is \(0\) then the necessary conditions for case one are met.
Therefore
\begin{align*} L &= [1] \end{align*}
Since \(r = {\frac {1}{4}}\) is not a function of \(x\), then there is no need run Kovacic algorithm to obtain a solution
for transformed ode \(z''=r z\) as one solution is
\[ z_1(x) = {\mathrm e}^{-\frac {x}{2}} \]
Using the above, the solution for the original ode can
now be found. The first solution to the original ode in \(y\) is found from
`Methodsfor second order ODEs:---Trying classification methods ---tryinga quadraturecheckingif the LODE has constant coefficients<-constant coefficients successful`