Hi. please help me solve the ffg. differential equation:
D^2y=3sinx-4y given y=0 and dy/dx=1 at x=pie/2
Need full solution. Tx
To solve the given differential equation, we can use the method of undetermined coefficients.
Step 1: Find the complementary solution:
Assume the solution is of the form y_c = Ae^rx, where A and r are constants to be determined. Taking the second derivative of y_c, we have D^2y_c = Ar^2e^rx.
Substituting this into the differential equation, we get:
Ar^2e^rx = 3sin(x) - 4Ae^rx
To equate the coefficients of like terms, we have:
Ar^2 = -4A (coefficient of e^rx terms)
0 = 3sin(x) (coefficient of sin(x) terms)
From the equation Ar^2 = -4A, we have two possibilities:
1) If r = 0, then A can be any constant.
2) If r ≠ 0, then A = 0.
So for the complementary solution, we have two cases:
1) y_c1 = Ae^0x = A (where A is a constant)
2) y_c2 = 0 (when r ≠ 0)
Therefore, the complementary solution is y_c = A, where A is a constant.
Step 2: Find the particular solution using the Method of Undetermined Coefficients:
Since the right-hand side of the equation is sin(x), we assume a particular solution of the form y_p = Bsin(x), where B is a constant to be determined.
Taking the first and second derivatives, we have:
dy_p/dx = Bcos(x)
D^2y_p = -Bsin(x)
Substituting into the differential equation, we get:
-Bsin(x) = 3sin(x) - 4(Bsin(x))
Equating the coefficients of like terms, we have:
-1 = 3 - 4B
Solving for B, we find that B = -1/4.
Therefore, the particular solution is y_p = (-1/4)sin(x).
Step 3: Find the general solution:
The general solution is given by y = y_c + y_p.
1) When r = 0:
y = A + (-1/4)sin(x)
2) When r ≠ 0:
y = 0 + (-1/4)sin(x)
So, the general solution is:
y = A + (-1/4)sin(x)
Step 4: Applying the initial conditions:
Given y(π/2) = 0 and dy/dx (π/2) = 1:
1) When r = 0:
At x = π/2: A - (1/4) = 0 --> A = 1/4
Thus, the solution in this case is:
y = (1/4) + (-1/4)sin(x)
2) When r ≠ 0:
At x = π/2: 0 - (1/4) = 0 (which is inconsistent)
Therefore, there is no solution in this case.
Thus, the final solution to the differential equation is:
y = (1/4) + (-1/4)sin(x)