To solve the differential equation (D2 + 4)y = tan2x using the variation of parameter method, we need to follow these steps:

Step 1: Find the complementary solution
First, we find the complementary solution by solving the homogeneous equation (D2 + 4)y = 0. The characteristic equation is r^2 + 4 = 0, which gives us the roots r = ±2i. Therefore, the complementary solution is y_c(x) = c1*cos(2x) + c2*sin(2x), where c1 and c2 are arbitrary constants.

Step 2: Find the particular solution
To find the particular solution, we assume that y_p(x) = u1(x)*cos(2x) + u2(x)*sin(2x), where u1(x) and u2(x) are functions to be determined.

Step 3: Find the derivatives of u1(x) and u2(x)
Differentiate y_p(x) with respect to x to find its first and second derivatives:
y_p'(x) = u1'(x)*cos(2x) - 2u1(x)*sin(2x) + u2'(x)*sin(2x) + 2u2(x)*cos(2x)
y_p''(x) = u1''(x)*cos(2x) - 4u1'(x)*sin(2x) - 4u1(x)*cos(2x) + u2''(x)*sin(2x) + 4u2'(x)*cos(2x) - 4u2(x)*sin(2x)

Step 4: Substitute y_p(x), y_p'(x), and y_p''(x) into the original equation
Substitute y_p(x), y_p'(x), and y_p''(x) into the original equation (D2 + 4)y = tan2x, and simplify the equation.

Step 5: Solve for u1'(x) and u2'(x)
Equate the coefficients of cos(2x) and sin(2x) on both sides of the equation obtained in step 4. This will give us two equations for u1'(x) and u2'(x).

Step 6: Integrate u1'(x) and u2'(x) to find u1(x) and u2(x)
Integrate the equations obtained in step 5 to find u1(x) and u2(x). Remember to include integration constants.

Step 7: Substitute u1(x) and u2(x) into y_p(x)
Substitute the values of u1(x) and u2(x) obtained in step 6 into the expression for y_p(x) to find the particular solution y_p(x).

Step 8: Find the general solution
The general solution of the differential equation (D2 + 4)y = tan2x is given by y(x) = y_c(x) + y_p(x), where y_c(x) is the complementary solution obtained in step 1 and y_p(x) is the particular solution obtained in step 7.

Note: The variation of parameter method is used to find the particular solution of a non-homogeneous linear differential equation. It involves assuming the particular solution in a specific form and then determining the unknown functions by substituting them into the original equation.

Question

To solve the differential equation (D2 + 4)y = tan2x using the variation of parameter method, we need to follow these steps:

Step 1: Find the complementary solution
First, we find the complementary solution by solving the homogeneous equation (D2 + 4)y = 0. The characteristic equation is r^2 + 4 = 0, which gives us the roots r = ±2i. Therefore, the complementary solution is y_c(x) = c1*cos(2x) + c2*sin(2x), where c1 and c2 are arbitrary constants.

Step 2: Find the particular solution
To find the particular solution, we assume that y_p(x) = u1(x)*cos(2x) + u2(x)*sin(2x), where u1(x) and u2(x) are functions to be determined.

Step 3: Find the derivatives of u1(x) and u2(x)
Differentiate y_p(x) with respect to x to find its first and second derivatives:
y_p'(x) = u1'(x)*cos(2x) - 2u1(x)*sin(2x) + u2'(x)*sin(2x) + 2u2(x)*cos(2x)
y_p''(x) = u1''(x)*cos(2x) - 4u1'(x)*sin(2x) - 4u1(x)*cos(2x) + u2''(x)*sin(2x) + 4u2'(x)*cos(2x) - 4u2(x)*sin(2x)

Step 4: Substitute y_p(x), y_p'(x), and y_p''(x) into the original equation
Substitute y_p(x), y_p'(x), and y_p''(x) into the original equation (D2 + 4)y = tan2x, and simplify the equation.

Step 5: Solve for u1'(x) and u2'(x)
Equate the coefficients of cos(2x) and sin(2x) on both sides of the equation obtained in step 4. This will give us two equations for u1'(x) and u2'(x).

Step 6: Integrate u1'(x) and u2'(x) to find u1(x) and u2(x)
Integrate the equations obtained in step 5 to find u1(x) and u2(x). Remember to include integration constants.

Step 7: Substitute u1(x) and u2(x) into y_p(x)
Substitute the values of u1(x) and u2(x) obtained in step 6 into the expression for y_p(x) to find the particular solution y_p(x).

Step 8: Find the general solution
The general solution of the differential equation (D2 + 4)y = tan2x is given by y(x) = y_c(x) + y_p(x), where y_c(x) is the complementary solution obtained in step 1 and y_p(x) is the particular solution obtained in step 7.

Note: The variation of parameter method is used to find the particular solution of a non-homogeneous linear differential equation. It involves assuming the particular solution in a specific form and then determining the unknown functions by substituting them into the original equation.

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