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Find the solution of y′′+2y′=64sin(2t)+32cos(2t) with y(0)=1 and u′(0)=8. y=

User Asad Malik
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Final answer:

To solve the given differential equation, use the method of undetermined coefficients to find a particular solution. Add the complementary solution to get the complete solution. Use the initial conditions to determine the values of the constants.

Step-by-step explanation:

To solve the given differential equation, we can use the method of undetermined coefficients. Since the right side of the equation contains both sine and cosine terms, we can assume a particular solution of the form y = Asin(2t) + Bcos(2t). We can then substitute this assumption into the differential equation and solve for A and B.

After solving the differential equation, we find that the particular solution is y = 4sin(2t) + 2cos(2t). To find the complete solution, we add the complementary solution which satisfies the homogeneous equation y'' + 2y' = 0. This complementary solution is given by y = C1e^(-t) + C2, where C1 and C2 are constants. Finally, we can use the initial conditions y(0) = 1 and y'(0) = 8 to determine the values of C1 and C2, and thus obtain the complete solution.

User Archagon
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