O.D.E. Complementary and Particular Solution

In summary, the problem is to find the values of C_1 and C_2 in the given ode, with initial conditions y(0) = 3 and y'(0) = 11. The solution involves using the complementary solution y_c and particular solution y_p to get the general solution of y(x). The correct values for C_1 and C_2 are 1 and 4, but the individual came up with 2 and 3 due to a simple arithmetic error.
  • #1
kahless2005
46
0
Not exactly homework, but it is a problem I'm having...

Im given an ode that reads:
y"-2y'-3y = 6;
[itex]y_c = C_1 * /exp^-x + C_2 * /exp^3x[/itex]
[itex]y_p[/itex] is -2

y(0) = 3
y'(0) = 11

Now I am tasked to find what [itex]C_1[/itex] and [itex]C_2[/itex] are.

I know that y(x) = [itex]y_c + y_p[/itex]
so:
y = [itex]C_1 * /exp^-x + C_2 * /exp^3x[/itex] - 2
and
y' = [itex]-C_1 * /exp^-x + 3 * C_2 * /exp^3x[/itex]

The book defines the answers as:
[itex]C_1[/itex] = 1 and [itex]C_2[/itex] = 4

Yet when I work it out, I've gotten [itex]C_1[/itex] = 2 and [itex]C_2[/itex] = 3.

What am I doing wrong?

NOTE: I hope I did the itex right... my computer isn't showing them... :biggrin:
 
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  • #2
Nevermind... I got it to work. I had a simple Aritmatic Error
 

FAQ: O.D.E. Complementary and Particular Solution

What is an O.D.E. Complementary Solution?

An O.D.E. (ordinary differential equation) complementary solution is the general solution to the homogeneous form of the differential equation. It is found by setting the right-hand side of the equation to zero and solving for the dependent variable.

How is a Particular Solution different from a Complementary Solution?

A particular solution is a specific solution to the non-homogeneous form of the differential equation, while a complementary solution is the general solution to the homogeneous form. The particular solution accounts for the specific initial/boundary conditions of the problem.

Why do we need both a Complementary Solution and a Particular Solution?

We need both a complementary solution and a particular solution because they work together to form the general solution to the non-homogeneous differential equation. The complementary solution accounts for the behavior of the system without any external inputs, while the particular solution accounts for the behavior caused by the external inputs.

How do you determine the constants in the Complementary Solution?

The constants in the complementary solution are determined by solving the homogeneous form of the differential equation using the initial/boundary conditions of the problem. These constants are then used to form the general solution.

Can you explain the concept of a Superposition of Solutions in O.D.E.?

The Superposition of Solutions in O.D.E. states that the general solution to a non-homogeneous equation can be expressed as the sum of the complementary solution and the particular solution. This means that the combined effect of both the homogeneous and non-homogeneous parts of the equation can be represented by the sum of their individual solutions.

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