157k views
2 votes
Find the solution of y′′+3y′=18sin(3t)+144cos(3t) with ϰ(n)=7 and ϰ′(n)=4 y

User Aritra B
by
7.8k points

1 Answer

6 votes

Final answer:

To solve the differential equation y′′+3y′=18sin(3t)+144cos(3t), you can use the method of undetermined coefficients and assume a particular solution of the form: y_p(t) = A sin(3t) + B cos(3t). By substituting this particular solution into the equation and comparing coefficients, you can find the values of A and B. The general solution to the differential equation is the sum of the homogeneous solution and the particular solution.

Step-by-step explanation:

To solve the differential equation y′′+3y′=18sin(3t)+144cos(3t), we can use the method of undetermined coefficients. Since the equation is non-homogeneous, we assume a particular solution of the form: y_p(t) = A sin(3t) + B cos(3t), where A and B are constants to be determined.

By substituting this particular solution into the equation and comparing coefficients, we find that A = -6 and B = 48. Therefore, the particular solution is y_p(t) = -6sin(3t) + 48cos(3t).

The general solution to the differential equation is the sum of the homogeneous solution and the particular solution: y(t) = y_h(t) + y_p(t). The homogeneous solution is y_h(t) = c1e^(-3t) + c2, where c1 and c2 are constants determined by initial conditions.

User Enith
by
7.5k points

Related questions

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.