Solve the boundary-value problem.
y'' + 5y' - 6y = 0 , y(0) = 0 , y(2) = 1
y = c1 e^-6x + c2 e^x
Plug in your boundary conditions, and you get
c1 + c2 = 0
c1/e^12 + c2/e^2 = 1
Now just solve for c1 and c2
To solve the given boundary-value problem, we need to find the function y(x) that satisfies the given differential equation and boundary conditions.
The differential equation is a second-order linear homogeneous ordinary differential equation. We can solve it using the characteristic equation.
Step 1: Find the characteristic equation:
The characteristic equation is obtained by assuming a solution of the form y = e^(rx), where r is a constant.
Plugging this into the differential equation, we get:
r^2 + 5r - 6 = 0
Step 2: Solve the characteristic equation:
We can factorize the equation: (r + 6)(r - 1) = 0
This yields two roots: r1 = -6 and r2 = 1.
Step 3: Determine the general solution:
The general solution of the differential equation is given by:
y(x) = C1 * e^(-6x) + C2 * e^x
Step 4: Apply the boundary conditions:
Using the first boundary condition, y(0) = 0, we substitute x = 0 into the general solution:
0 = C1 * e^(0) + C2 * e^(0)
0 = C1 + C2
Using the second boundary condition, y(2) = 1, we substitute x = 2 into the general solution:
1 = C1 * e^(-12) + C2 * e^(2)
Step 5: Solve the system of equations:
We have two equations with two unknowns: C1 and C2.
Solving the system of equations:
0 = C1 + C2
1 = C1 * e^(-12) + C2 * e^(2)
Using the first equation, we can express C2 in terms of C1:
C2 = -C1
Substituting this into the second equation:
1 = C1 * e^(-12) - C1 * e^(2)
Simplifying further:
1 = C1 * (e^(-12) - e^(2))
Solving for C1:
C1 = 1 / (e^(-12) - e^(2))
Substituting C1 back into the first equation:
C2 = -C1
So, we have found the values of C1 and C2:
C1 = 1 / (e^(-12) - e^(2))
C2 = -1 / (e^(-12) - e^(2))
Step 6: Write the final solution:
Now that we have the values of C1 and C2, we can write the final solution to the boundary-value problem:
y(x) = (1 / (e^(-12) - e^(2))) * e^(-6x) - (1 / (e^(-12) - e^(2))) * e^x