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For the following differential equation

d²y/dt²+4y= 1/cos(2t) +7t+1
a. Find the general solution
b. Find a solution such that y(0)=0,y (0)=1.
Justify your answers.

User Lamond
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Final answer:

The general solution to the given differential equation can be found by solving the homogeneous equation and guessing a particular solution. The initial conditions y(0) = 0 and y'(0) = 1 are used to determine the unknown constants in the solution.

Step-by-step explanation:

To solve the differential equation d²y/dt² + 4y = 1/cos(2t) + 7t + 1, we can find the general solution by considering the homogeneous part and the particular solution separately.

General Solution of the Homogeneous Equation

The homogeneous equation is d²y/dt² + 4y = 0. The characteristic equation of this second order differential equation is r² + 4 = 0. Solving this, we get r = ±2i, so the general solution to the homogeneous equation is y_h(t) = C_1 cos(2t) + C_2 sin(2t).

Particular Solution

Next, we guess a particular solution of the form y_p(t) = A/cos(2t) + Bt + C. Plugging this into the non-homogeneous equation and solving for A, B, and C will give us the coefficients for the particular solution.

Initial Conditions

With the initial conditions y(0) = 0 and y'(0) = 1, we can plug the general solution and its derivative into these conditions to solve for the constants C_1 and C_2.

The complete general solution of the given differential equation will be a combination of the homogeneous solution and the particular solution: y(t) = y_h(t) + y_p(t). Once the constants A, B, C, C_1, and C_2 are found, we'll have the function that satisfies the differential equation along with the initial conditions.

User Jimbo Jonny
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