Internal
problem
ID
[8050]
Book
:
Second
order
enumerated
odes
Section
:
section
1
Problem
number
:
13
Date
solved
:
Monday, October 21, 2024 at 04:44:49 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
Solve
Time used: 0.536 (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.
Methods for second order ODEs:
Solving time : 0.059
(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]