d2y/dx2+9y=cos2x+sin2x
To solve the differential equation d^2y/dx^2 + 9y = cos(2x) + sin(2x), we will first find the complementary solution and then the particular solution.
Step 1: Find the complementary solution:
The characteristic equation for the given differential equation is r^2 + 9 = 0. Solving this quadratic equation, we get two complex roots: r = ± 3i.
The complementary solution is then of the form y_c(x) = A*cos(3x) + B*sin(3x), where A and B are arbitrary constants.
Step 2: Find the particular solution:
To find the particular solution, we need to consider the non-homogeneous part of the equation, which is cos(2x) + sin(2x).
We assume the particular solution to be of the form y_p(x) = C*cos(2x) + D*sin(2x), where C and D are constants to be determined.
Taking the first and second derivatives of y_p(x), we have:
y'_p(x) = -2C*sin(2x) + 2D*cos(2x)
y''_p(x) = -4C*cos(2x) - 4D*sin(2x)
Substituting these derivatives and y_p(x) into the original differential equation, we get:
(-4C*cos(2x) - 4D*sin(2x)) + 9(C*cos(2x) + D*sin(2x)) = cos(2x) + sin(2x)
Simplifying this equation, we have:
(9C - 4C)*cos(2x) + (9D - 4D)*sin(2x) = cos(2x) + sin(2x)
(5C)*cos(2x) + (5D)*sin(2x) = cos(2x) + sin(2x)
Comparing the coefficients on both sides, we have:
5C = 1
5D = 1
Solving these equations, we get C = 1/5 and D = 1/5.
Therefore, the particular solution is: y_p(x) = (1/5)*cos(2x) + (1/5)*sin(2x).
Step 3: Find the general solution:
The general solution is the sum of the complementary solution and the particular solution:
y(x) = y_c(x) + y_p(x)
y(x) = A*cos(3x) + B*sin(3x) + (1/5)*cos(2x) + (1/5)*sin(2x)
So, the general solution to the given differential equation is:
y(x) = A*cos(3x) + B*sin(3x) + (1/5)*cos(2x) + (1/5)*sin(2x)
To solve the differential equation d^2y/dx^2 + 9y = cos(2x) + sin(2x), you can follow these steps:
Step 1: Find the homogeneous solution.
To find the homogeneous solution, set the right-hand side of the equation (cos(2x) + sin(2x)) equal to zero. The equation becomes:
d^2y/dx^2 + 9y = 0
This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is:
r^2 + 9 = 0
Solving the characteristic equation, we find two complex roots: r = ±3i
The homogeneous solution is then in the form:
y_h(x) = C1*cos(3x) + C2*sin(3x)
Step 2: Find a particular solution.
To find a particular solution, we need to consider the non-homogeneous part of the equation: cos(2x) + sin(2x).
Since cos(2x) and sin(2x) are both periodic functions with a period of π, we can assume a particular solution of the form:
y_p(x) = A*cos(2x) + B*sin(2x)
Differentiating y_p(x) twice with respect to x, we have:
d^2y_p/dx^2 = -4A*cos(2x) - 4B*sin(2x)
Substituting these into the original differential equation, we get:
(-4A*cos(2x) - 4B*sin(2x)) + 9(A*cos(2x) + B*sin(2x)) = cos(2x) + sin(2x)
By equating the coefficients of sin(2x) and cos(2x) separately, we have:
-4A + 9A = 0 -> 5A = 0
-4B + 9B = 1 -> 5B = 1
Solving these equations gives A = 0 and B = 1/5. Therefore, our particular solution is:
y_p(x) = (1/5)*sin(2x)
Step 3: Find the general solution.
The general solution for the given differential equation is the sum of the homogeneous solution (y_h(x)) and the particular solution (y_p(x)):
y(x) = C1*cos(3x) + C2*sin(3x) + (1/5)*sin(2x)
Where C1 and C2 are arbitrary constants determined by any initial or boundary conditions.
That's it! You've solved the differential equation d^2y/dx^2 + 9y = cos(2x) + sin(2x).
y+9y=3x+e^x
y" + 9y = cos2x+sin2x
You know that
y"+9y=0
has solutions
y = a cos3x + b sin3x
so the solution here is
y = a cos 3x + b sin 3x + (cos2x+sin2x)/5