Solving a System of Differential Equations with Initial Condition

Published on May 29, 2025

Calculus

Find the solution to a system of differential equations given its general solution and an initial condition. The solution involves finding the constants and then substituting them back into the general solution to find the specific solution for x1(t), x2(t), and x3(t).

Question

Answer

The problem asks us to find

, and

given the general solution for the system

and an initial condition.

The general solution is given as:

The initial condition is given as

First, let’s substitute

into the general solution to find the values of

, and

.
When

, and

.
So,

We are given

.
Therefore, we have the following system of linear equations:

This can be written as:

Let’s simplify the system:

From equation (1),

.
Substitute this into equation (3):

Now we have a system of two equations with

and

:
A.

Subtract equation A from equation B:

Now substitute

into equation A:

Finally, substitute

into the expression for

So, we have

, and

Now substitute these values back into the general solution:

Therefore, the components are:

The final answer is

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