Questions: Find the solution to the boundary value problem: d^2 y/d t^2 - 8 d y/d t + 12 y = 0, y(0) = 8, y(1) = 7 y=

Transcript text: Find the solution to the boundary value problem: \[ \begin{array}{l} \frac{d^{2} y}{d t^{2}}-8 \frac{d y}{d t}+12 y=0, \quad y(0)=8, y(1)=7 \\ y= \end{array} \]

Solution

Solution Steps

To solve the boundary value problem, we need to:

Solve the characteristic equation of the differential equation to find the general solution.
Apply the boundary conditions to determine the specific constants in the general solution.

Step 1: Solve the Characteristic Equation

The given differential equation is: \[ \frac{d^{2} y}{d t^{2}} - 8 \frac{d y}{d t} + 12 y = 0 \] The characteristic equation for this differential equation is: \[ r^2 - 8r + 12 = 0 \] Solving this quadratic equation, we get the roots: \[ r = 4 \quad \text{and} \quad r = 2 \]

Step 2: Form the General Solution

Using the roots of the characteristic equation, the general solution of the differential equation is: \[ y(t) = (C_1 + C_2 e^{4t}) e^{2t} \]

Step 3: Apply Boundary Conditions

We are given the boundary conditions: \[ y(0) = 8 \quad \text{and} \quad y(1) = 7 \] Substituting \( t = 0 \) into the general solution: \[ y(0) = (C_1 + C_2 e^{0}) e^{0} = C_1 + C_2 = 8 \] Substituting \( t = 1 \) into the general solution: \[ y(1) = (C_1 + C_2 e^{4}) e^{2} = C_1 e^{2} + C_2 e^{6} = 7 \]

Step 4: Solve for Constants

We solve the system of equations: \[ \begin{cases} C_1 + C_2 = 8 \\ C_1 e^{2} + C_2 e^{6} = 7 \end{cases} \] Solving these equations, we get: \[ C_1 = \frac{-7 + 8 e^{6}}{-e^{2} + e^{6}} \quad \text{and} \quad C_2 = \frac{7 - 8 e^{2}}{-e^{2} + e^{6}} \]

Step 5: Form the Particular Solution

Substituting the values of \( C_1 \) and \( C_2 \) back into the general solution, we get: \[ y(t) = \left( \frac{7 - 8 e^{2}}{-e^{2} + e^{6}} e^{4t} + \frac{-7 + 8 e^{6}}{-e^{2} + e^{6}} \right) e^{2t} \]