Solving Initial Value Problems Involving Differential Equations with Repeated Roots
The initial value problem in differential equations is a common problem that arises in various fields such as physics, engineering, and economics. The problem involves finding a function that satisfies a given differential equation along with specific initial conditions. This article focuses on solving a particular type of differential equation with a repeated root, and provides a step-by-step guide to solving the initial value problem given by the equation y'' - 2y' y 0. The initial conditions are y(0) 3 and y'(0) 1.
Understanding the Problem and the Method
To solve an initial value problem, we need to follow several steps. First, we will identify the characteristic equation and solve for the roots. Then, we will derive the general solution of the differential equation. Finally, we will apply the initial conditions to find the specific solution.
Step 1: Find the Characteristic Equation
We start by assuming a solution of the form (y e^{rt}). Substituting this into the differential equation y'' - 2y' y 0 gives us the characteristic equation:
[r^2 - 2r 1 0]This is a quadratic equation that can be factored:
[(r - 1)^2 0]The characteristic equation has a repeated root, (r 1).
Step 2: Solve the Characteristic Equation
Since the characteristic equation has a repeated root, the general solution to the differential equation is given by:
[y(t) C_1e^t C_2te^t]Step 3: Apply Initial Conditions
To determine the unique solution, we need to apply the initial conditions. The first condition is (y(0) 3):
[y(0) C_1e^0 C_2(0)e^0 C_1 3]So, we have:
[C_1 3]For the second condition, (y'(0) 1), we first need to find the derivative of the general solution:
[y(t) 3e^t C_2te^t] [y'(t) 3e^t C_2(e^t te^t) 3e^t C_2e^t(1 t)]Evaluating at (t 0):
[y'(0) 3e^0 C_2e^0(1 0) 3 C_2 1]Solving for (C_2):
[C_2 1 - 3 -2]Step 4: Write the Final Solution
Now that we have determined the constants (C_1) and (C_2), we can write the solution to the initial value problem:
[[y(t) 3e^t - 2te^t]]Conclusion
The solution to the initial value problem is given by:
[boxed{y(t) 3e^t - 2te^t}]