solve y"+4y-4z'=3x
z"+z-y'=1-x
where y(0)=z(0)=1
and y'(0)=z'(0)=0
by laplace transform
no ideas?
Recall that
L{y} = F(s)
L{y'} = s F(s) - f(0)
L{y"} = s^2 F(s) - s f(0) - f'(0)
then just collect terms, solve for s, and do L^-1{s}
To solve the given system of differential equations using Laplace transform, we need to take Laplace transforms of both sides of each equation. The Laplace transform of a function f(t) is denoted as F(s) and is defined as follows:
L{f(t)} = F(s) = ∫[0 to ∞] e^(-st) * f(t) dt
Let's take the Laplace transform of each equation step by step.
1. Taking the Laplace transform of the first equation:
L{y''} + 4L{y} - 4sL{z'} = 3L{x}
Using the property L{f'} = sF(s) - f(0), where f(0) is the initial condition, we can rewrite the equation as follows:
s^2Y(s) - sy(0) - y'(0) + 4Y(s) - 4sZ(s) + 4z(0) - 3X(s) = 0
Substituting the initial conditions y(0) = 1, y'(0) = 0, and rearranging the terms, we get:
s^2Y(s) + 4Y(s) - 4sZ(s) - 3X(s) = s - 4
2. Taking the Laplace transform of the second equation:
L{z''} + L{z} - L{y'} = 1 - L{x}
Using the property L{f'} = sF(s) - f(0) and substituting the initial condition z(0) = 1, z'(0) = 0, we have:
s^2Z(s) - sz(0) - z'(0) + Z(s) - sY(s) = 1 - X(s)
Rearranging the terms, we get:
s^2Z(s) + Z(s) - sY(s) - X(s) = s + 1
3. Now, we have a system of equations in terms of Laplace transforms. Let's solve for Y(s) and Z(s) by eliminating X(s) between the two equations.
Multiply the first equation by (s + 1) and the second equation by (-4):
(s^2 + 4s + 3)Y(s) - (4s^2 - 1)Z(s) + 4(s + 1) = 4s + 4
(-4s^2 + s + 3)Y(s) + (s^2 + s - 4)Z(s) = -4s - 4
Adding these two equations together, we can eliminate X(s):
2sY(s) - 2sZ(s) = 0
Simplifying, we get:
sY(s) = sZ(s)
4. Now, let's solve for Y(s) and Z(s) in terms of s:
Divide the second equation by (s^2 + s - 4):
Y(s) = (4s + 4) / (-4s^2 + s + 3)
Z(s) = (-4s - 4) / (s^2 + s - 4)
5. Finally, let's take the inverse Laplace transform of Y(s) and Z(s) to find the solutions y(t) and z(t) respectively.
To simplify the partial fraction decomposition of Y(s) and Z(s), you may use a mathematical software or table of Laplace transform pairs to find the inverse Laplace transform.
By taking the inverse Laplace transform, you will obtain the solutions y(t) and z(t) for the given system of differential equations.
Note that this process can be complex and time-consuming. It may be helpful to use mathematical software like MATLAB or Wolfram Alpha to perform these calculations.