Internal
problem
ID
[9048]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
64
Date
solved
:
Wednesday, March 05, 2025 at 07:18:07 AM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
Solve
Time used: 0.567 (sec)
Let
Then
Therefore
Hence the given ode can now be written as
This is separable first order ode. Integrating
Replacing
Summary of solutions found
Time used: 2.763 (sec)
Writing the ode as
The condition of Lie symmetry is the linearized PDE given by
To determine
Where the unknown coefficients are
Substituting equations (1E,2E) and
Putting the above in normal form gives
Setting the numerator to zero gives
Since the PDE has radicals, simplifying gives
Looking at the above PDE shows the following are all the terms with
The following
substitution is now made to be able to collect on all terms with
The above PDE (6E) now becomes
Collecting the above on the terms
Equation (7E) now becomes
Setting each coefficients in (8E) to zero gives the following equations to solve
Solving the above equations for the unknowns gives
Substituting the above solution in the anstaz (1E,2E) (using
The next step is to determine the canonical coordinates
The characteristic pde which is used to find the canonical coordinates is
The above comes from the requirements that
This is easily solved to give
Where now the coordinate
And
Integrating gives
Where the constant of integration is set to zero as we just need one solution. Now that
Where in the above
Evaluating all the partial derivatives gives
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates.
We now need to express the RHS as function of
The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates
Since the ode has the form
This results in
Summary of solutions found
Time used: 13.437 (sec)
Let
Solving for
This has the form
Where
Solving ode 1A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for
Substituing the above solution for
Solving ode 2A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for
Substituing the above solution for
Solving ode 3A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
No valid singular solutions found.
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for
Substituing the above solution for
Solving ode 4A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
No valid singular solutions found.
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for
Substituing the above solution for
Solving ode 5A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
No valid singular solutions found.
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for
Substituing the above solution for
Solving ode 6A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for
Substituing the above solution for
Solving ode 7A
Taking derivative of (*) w.r.t.
Comparing the form
Hence (2) becomes
The singular solution is found by setting
Solving the above for
Substituting these in (1A) and keeping singular solution that verifies the ode gives
The general solution is found when
This ODE is now solved for
Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as
Singular solutions are found by solving
for
Substituing the above solution for
The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed. The solution
was found not to satisfy the ode or the IC. Hence it is removed.
Summary of solutions found
ode:=diff(y(x),x) = (Pi+x+7*y(x))^(7/2); dsolve(ode,y(x), singsol=all);
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying homogeneous C 1st order, trying the canonical coordinates of the invariance group -> Calling odsolve with the ODE`, diff(y(x), x) = -1/7, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- 1st order, canonical coordinates successful <- homogeneous successful`
Maple step by step
ode=D[y[x],x]==(Pi+x+7*y[x])^(7/2); ic={}; DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
from sympy import * x = symbols("x") y = Function("y") ode = Eq(-(x + 7*y(x) + pi)**(7/2) + Derivative(y(x), x),0) ics = {} dsolve(ode,func=y(x),ics=ics)