The initial velocity is:
u'(0) = [ax(0) + by(0), cx(0) + dy(0)]^T
Here's the solution,
1. Expressing the system in matrix form:
Create a vector of dependent variables:
u = [x, y]^T
Write the system as a matrix equation:
u' = Au
where:
A = [[a, b], [c, d]]
2. Calculating the initial velocity:
Plug in the initial values:
u'(0) = A * u(0) = [[a, b], [c, d]] * [[x(0)], [y(0)]] = [[ax(0) + by(0)], [cx(0) + dy(0)]]
Therefore, the initial velocity is:
u'(0) = [ax(0) + by(0), cx(0) + dy(0)]^T
Summary of steps:
- Identify the dependent variables and create a vector u.
- Write the differential equations in the form u' = Au.
- Form the coefficient matrix A.
- Plug in the given initial values u(0) into the matrix equation.
- Solve for u'(0) to obtain the initial velocity.
The probable question can be:
"Consider a dynamic system described by the following set of coupled first-order differential equations:
\[ \frac{dx}{dt} = ax + by \]
\[ \frac{dy}{dt} = cx + dy \]
Express the given system in matrix form. Given the initial values \(x(0)\) and \(y(0)\), determine the initial velocity of the solution. Clearly outline the steps involved in obtaining the solution."