d2y/dx2+9y=cos2x+sin2x
Sir solve this question
y" + 9y = 0
The homogeneous DE has solution y = c1 sin(3x) + c2 cos(3x)
So your equation has solution
y = c1 sin(3x) + c2 cos(3x) + 1/5 (sin2x + cos2x)
The given equation is a second-order linear homogeneous differential equation. To solve it, we need to find the general solution for y(x).
To solve this differential equation, we'll follow these steps:
Step 1: Find the complementary function
Step 2: Find the particular solution
Step 1: Find the complementary function:
The complementary function involves finding the solution to the homogeneous equation d^2y/dx^2 + 9y = 0. The characteristic equation for this homogeneous equation is r^2 + 9 = 0.
Solving the characteristic equation, we get:
r^2 = -9
r = ± √(-9)
r = ±3i
The complementary function is y_c = C1cos(3x) + C2sin(3x), where C1 and C2 are arbitrary constants.
Step 2: Find the particular solution:
To find the particular solution for y(x), we look at the right-hand side of the given equation (cos^2(x) + sin^2(x)).
Since cos^2(x) + sin^2(x) = 1, we can rewrite the equation as:
d^2y/dx^2 + 9y = 1
To find a particular solution, we assume y_p = A (a constant).
Taking the second derivative of y_p:
d^2y_p/dx^2 = 0
Plugging the assumed particular solution and its derivatives back into the differential equation:
0 + 9A = 1
9A = 1
A = 1/9
Therefore, the particular solution is y_p = 1/9.
Step 3: Find the general solution:
The general solution is the sum of the complementary function (y_c) and the particular solution (y_p).
y(x) = y_c + y_p
y(x) = C1cos(3x) + C2sin(3x) + 1/9
So, the general solution to the given differential equation is y(x) = C1cos(3x) + C2sin(3x) + 1/9.