Solve using the annihilator method Y"-8y'+20y=5xe^(4x) sin (2x)
To solve the given differential equation using the annihilator method, follow these steps:
Step 1: Find the homogeneous solution
- Determine the characteristic equation by substituting Y = e^(rt) into the homogeneous form of the differential equation: Y'' - 8Y' + 20Y = 0.
- The characteristic equation becomes r^2 - 8r + 20 = 0.
- Solve the characteristic equation for its roots (r1 and r2) using the quadratic formula or factoring: r1 = 4 + 3i and r2 = 4 - 3i.
- The homogeneous solution is in the form Yh(t) = C1e^(r1t) + C2e^(r2t), where C1 and C2 are constants.
Step 2: Find the particular solution
- Since the given differential equation contains terms with e^(4x) as well as the product of x, e^(4x), and sin (2x), we need to find the particular solution using the method of undetermined coefficients.
- Assume the particular solution has the form Yp(t) = A x e^(4x) sin (2x), where A is a constant to be determined.
- Differentiate Yp(t) twice with respect to t and substitute it into the differential equation Y'' - 8Y' + 20Y = 5xe^(4x) sin (2x).
- Equate the coefficients of the terms on both sides of the equation and solve for A.
Step 3: Combine the homogeneous and particular solutions
- The general solution to the differential equation is given by Y(t) = Yh(t) + Yp(t). Substitute in the values obtained in steps 1 and 2.
- Y(t) = (C1e^(4t) + C2e^(4t)) + (A x e^(4x) sin (2x)).
Step 4: Apply initial conditions (if given)
- If initial conditions are provided, substitute them into the general solution from step 3 and solve for the constants C1, C2, and A.
That's it! You have now solved the given differential equation using the annihilator method.
To solve the given differential equation using the annihilator method, we will follow these steps:
Step 1: Find the complementary solution
a) Solve the characteristic equation:
The characteristic equation for the differential equation is obtained by replacing the y terms with their corresponding derivatives:
r^2 - 8r + 20 = 0
Solving this quadratic equation gives us two distinct roots: r1 = 4 and r2 = 5.
Therefore, the complementary solution is given by:
Yc = C1e^(4x) + C2e^(5x), where C1 and C2 are arbitrary constants.
Step 2: Find the particular solution
a) Identify the particular form:
The differential equation includes the term 5xe^(4x) sin(2x), which can be divided into two parts:
- 5x, which is a polynomial of degree 1.
- e^(4x) sin(2x), which is a product of exponential and trigonometric functions.
Since the exponential function e^(4x) is already present in the complementary solution, we will need to multiply the particular solution by x to avoid redundancy. Therefore, we assume a particular solution of the form:
Yp = x(Ax + B)e^(4x) sin(2x)
Now, we need to find the values of A and B.
b) Differentiate and substitute:
Differentiate Yp twice with respect to x, using the product rule where necessary, and substitute the derivatives into the differential equation:
Yp' = (A + 2Ax + B)e^(4x) sin(2x) + 2(Ax + B)e^(4x) cos(2x)
Yp'' = (2A + 4(Ax + B) + 4A)e^(4x) sin(2x) + 4(Ax + B)e^(4x) cos(2x) + 4(Ax + B)e^(4x) cos(2x) - 4(Ax + B)e^(4x) sin(2x)
Substituting these into the given differential equation, we get:
[(2A + 4(Ax + B) + 4A) - 8((A + 2Ax + B)e^(4x) sin(2x) + 2(Ax + B)e^(4x) cos(2x)) + 20(Ax + B)e^(4x) sin(2x)] = 5xe^(4x) sin(2x)
Simplifying and equating coefficients of like terms, we obtain a system of equations for A and B.
c) Solve for A and B:
Solve the system of equations to find the values of A and B that satisfy the equation.
Step 3: Find the general solution
Combine the complementary solution and the particular solution to obtain the general solution:
Y = Yc + Yp = C1e^(4x) + C2e^(5x) + x(Ax + B)e^(4x) sin(2x)
That's it! You have now solved the given differential equation using the annihilator method.