Internal
problem
ID
[8496]
Book
:
Second
order
enumerated
odes
Section
:
section
1
Problem
number
:
13
Date
solved
:
Sunday, November 10, 2024 at 03:55:07 AM
CAS
classification
:
[[_2nd_order, _missing_x]]
Solve
Time used: 0.568 (sec)
This is second order ode with missing dependent variable \(y\). Let
Then
Hence the ode becomes
Which is now solve for \(p(x)\) as first order ode. Solving for the derivative gives these ODE’s to solve
Now each of the above is solved separately.
Solving Eq. (1)
Integrating gives
Singular solutions are found by solving
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.
Solving Eq. (2)
Integrating gives
Singular solutions are found by solving
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.
For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
For solution (3) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
Will add steps showing solving for IC soon.
Summary of solutions found
`Methods for second order ODEs: *** Sublevel 2 *** Methods for second order ODEs: Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(diff(diff(y(x), x), x), x)+1/2, y(x)` *** Sublevel 4 *** Methods for third order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful <- 2nd order ODE linearizable_by_differentiation successful ------------------- * Tackling next ODE. *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation <- 2nd order ODE linearizable_by_differentiation successful`
Solving time : 0.305
(sec)
Leaf size : 27
dsolve(diff(diff(y(x),x),x)^2+diff(y(x),x) = 0, y(x),singsol=all)
Solving time : 0.022
(sec)
Leaf size : 69
DSolve[{(D[y[x],{x,2}])^2+D[y[x],x]==0,{}}, y[x],x,IncludeSingularSolutions->True]