d^2y/dx^2+4dy/dx+4y=4cosx+3sinx
y = c1*e^-2x + c2*x*e^-2x + sin(x)
To solve the given differential equation:
Step 1: Identify the homogeneous equation.
The homogeneous equation is obtained by setting the right-hand side (4cos(x) + 3sin(x)) to zero:
d^2y/dx^2 + 4(dy/dx) + 4y = 0
Step 2: Solve the homogeneous equation.
To solve the homogeneous equation, assume a solution of the form y = e^(mx). Substitute this into the equation to obtain the characteristic equation:
m^2 + 4m + 4 = 0
Step 3: Solve the characteristic equation.
The characteristic equation can be factored as (m+2)^2 = 0, which gives m = -2 (double root).
Step 4: Write the homogeneous solution.
The homogeneous solution is given by y_h = c1e^(-2x) + c2xe^(-2x), where c1 and c2 are constants.
Step 5: Find a particular solution.
To find a particular solution to the non-homogeneous equation, use the method of undetermined coefficients. Since the right-hand side contains both cos(x) and sin(x), assume a particular solution of the form:
y_p = Acos(x) + Bsin(x)
Step 6: Differentiate the particular solution.
Differentiate the particular solution to calculate dy_p/dx and d^2y_p/dx^2.
dy_p/dx = -Asin(x) + Bcos(x)
d^2y_p/dx^2 = -Acos(x) - Bsin(x)
Step 7: Substitute the particular solution into the original equation.
Substitute the particular solution, its first derivative, and its second derivative into the original differential equation:
(-Acos(x) - Bsin(x)) + 4(-Asin(x) + Bcos(x)) + 4(Acos(x) + Bsin(x)) = 4cos(x) + 3sin(x)
Step 8: Collect like terms.
Group the terms with the same trigonometric functions together:
(-3A + 4B)cos(x) + (-3B - 4A)sin(x) = 4cos(x) + 3sin(x)
Step 9: Set coefficients of trigonometric terms equal.
Equating the coefficients of cos(x) and sin(x) gives the following equations:
-3A + 4B = 4
-3B - 4A = 3
Step 10: Solve the system of equations.
Solving the system of equations gives A = -5/8 and B = 7/8.
Step 11: Write the particular solution.
Substituting the values of A and B back into the assumed particular solution gives:
y_p = (-5/8)cos(x) + (7/8)sin(x)
Step 12: Write the general solution.
The general solution to the non-homogeneous equation is given by combining the homogeneous solution and the particular solution:
y = y_h + y_p = c1e^(-2x) + c2xe^(-2x) + (-5/8)cos(x) + (7/8)sin(x)
This is the final solution to the given differential equation.
The given differential equation is a second-order linear homogeneous ordinary differential equation.
To solve it, we can follow the following steps:
Step 1: Find the auxiliary equation
The auxiliary equation is obtained by replacing the derivatives of y in the differential equation with the corresponding terms in the auxiliary equation:
r^2 + 4r + 4 = 0
Step 2: Solve the auxiliary equation
We can solve the quadratic equation to find the roots (values of r):
(r + 2)^2 = 0
r = -2
So, the roots of the auxiliary equation are r = -2, -2.
Step 3: Write the general solution
Since the roots are repeated, the general solution can be written as:
y = (C1 + C2*x)*e^(-2x), where C1 and C2 are constants.
Step 4: Find particular solution
To find a particular solution to the non-homogeneous part of the equation, we can use the method of undetermined coefficients. In this case, we need to find a particular solution to the equation 4cos(x) + 3sin(x).
Assuming a particular solution of the form:
yp(x) = A*cos(x) + B*sin(x)
where A and B are unknown constants.
Taking the derivatives and substituting them into the differential equation, we can solve for A and B.
Step 5: Write the complete solution
The complete solution is the sum of the general solution (step 3) and the particular solution (step 4):
y = (C1 + C2*x)*e^(-2x) + A*cos(x) + B*sin(x)
So, this is the general solution to the given differential equation.