Internal
problem
ID
[8859]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
1
Date
solved
:
Wednesday, March 05, 2025 at 06:56:29 AM
CAS
classification
:
[[_2nd_order, _missing_x]]
Solve
Time used: 0.455 (sec)
This is second order with constant coefficients homogeneous ODE. In standard form the ODE is
Where in the above
Since exponential function is never zero, then dividing Eq(2) throughout by
Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form.Using the quadratic formula
Substituting
Hence
Which simplifies to
The roots are complex but they are not conjugate of each others. Hence simplification using Euler relation is not possible here. Therefore the final solution is
Will add steps showing solving for IC soon.
Summary of solutions found
Time used: 0.170 (sec)
Writing the ode as
Comparing (1) and (2) shows that
Applying the Liouville transformation on the dependent variable gives
Then (2) becomes
Where
Substituting the values of
Comparing the above to (5) shows that
Therefore eq. (4) becomes
Equation (7) is now solved. After finding
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
Case |
Allowed pole order for |
Allowed value for |
1 |
|
|
2 |
Need to have at least one pole
that is either order |
no condition |
3 |
|
|
The order of
There are no poles in
Since
Using the above, the solution for the original ode can now be found. The first solution to the original ode in
Which simplifies to
The second solution
Substituting gives
Therefore the solution is
Will add steps showing solving for IC soon.
Summary of solutions found
ode:=diff(diff(y(x),x),x)+c*diff(y(x),x)+k*y(x) = 0; dsolve(ode,y(x), singsol=all);
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
Maple step by step
ode=D[y[x],{x,3}]-x^3*D[y[x],x]-x^2*y[x]-x^3==0; ic={}; DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
from sympy import * x = symbols("x") c = symbols("c") k = symbols("k") y = Function("y") ode = Eq(c*Derivative(y(x), x) + k*y(x) + Derivative(y(x), (x, 2)),0) ics = {} dsolve(ode,func=y(x),ics=ics)