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Find the general solution of the differential equation

y" - 36y = -108t + 72t^2.
NOTE: Use t as the independent variable. Use c_1 and c_2 as arbitrary constants. y(t): =________________

User Samiya
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1 Answer

5 votes

Answer:

y(t) = c_1e^(6t) + c_2e^(-6t) - 2t^2 + 3t,

Explanation:

To find the general solution of the given differential equation, we can first solve the associated homogeneous equation, and then find a particular solution for the non-homogeneous equation. Let's proceed with the steps:

Step 1: Solve the associated homogeneous equation:

The associated homogeneous equation is obtained by setting the right-hand side of the differential equation to zero:

y" - 36y = 0

The characteristic equation for this homogeneous equation is:

r^2 - 36 = 0

Solving the characteristic equation, we get the roots:

r = ±6

Therefore, the homogeneous solution is given by:

y_h(t) = c_1e^(6t) + c_2e^(-6t)

Step 2: Find a particular solution for the non-homogeneous equation:

We can use the method of undetermined coefficients to find a particular solution for the non-homogeneous equation. Since the right-hand side of the equation is a polynomial, we assume a particular solution of the form:

y_p(t) = At^2 + Bt + C

Now we can substitute this particular solution into the original differential equation and solve for the coefficients A, B, and C.

y_p"(t) - 36y_p(t) = -108t + 72t^2

Differentiating y_p(t) twice:

y_p'(t) = 2At + B

y_p"(t) = 2A

Substituting into the differential equation:

2A - 36(At^2 + Bt + C) = -108t + 72t^2

Simplifying and equating coefficients:

-36A = 72 (coefficient of t^2)

-36B = -108t (coefficient of t)

-36C = 0 (coefficient of the constant term)

Solving these equations, we find:

A = -2

B = 3

C = 0

So the particular solution is:

y_p(t) = -2t^2 + 3t

Step 3: Write the general solution:

The general solution of the non-homogeneous equation is the sum of the homogeneous and particular solutions:

y(t) = y_h(t) + y_p(t)

= c_1e^(6t) + c_2e^(-6t) - 2t^2 + 3t

Therefore, the general solution of the given differential equation is:

y(t) = c_1e^(6t) + c_2e^(-6t) - 2t^2 + 3t,

where c_1 and c_2 are arbitrary constants.

User Svintus
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