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[NEEDITNOW0819]

D³y(t)+9D²y(t)+25Dy(t)+17y(t)= 5 cos(3t); D²y(0)=0 , Dy(0)=1, y(0)=0
determine the total solution using Laplace Transform and Classical Method

Need correct solutions and explanation

 

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Laplace Transform Method:

Taking the Laplace Transform of both sides of the equation, we get:

L{D³y(t)} + 9L{D²y(t)} + 25L{Dy(t)} + 17L{y(t)} = 5L{cos(3t)}

Using the properties of Laplace Transform, we get:

s³Y(s) - s²y(0) - sy'(0) - y''(0) + 9s²Y(s) - 9sy(0) - 9y'(0) + 25sY(s) - 25y(0) + 17Y(s) = 5[s/(s²+9)]

Substituting the initial conditions, we get:

s³Y(s) + 9s²Y(s) + 25sY(s) + 17Y(s) = 5s/(s²+9) + s²

Simplifying, we get:

Y(s) = [5s + s²(s²+9)]/[(s²+9)(s³+9s²+25s+17)]

Using partial fraction decomposition, we get:

Y(s) = -1/(s+3) + 2/(s²+9) + (3s-2)/(s²+4s+13) - (s+1)/(s²+4s+13) - 1/(s-1)

Taking the inverse Laplace Transform, we get the total solution:

y(t) = -e^(-3t) + 2sin(3t) + (3/2)e^(-2t)sin(t) - (1/2)e^(-2t)cos(t) - e^t

Classical Method:

The characteristic equation of the differential equation is:

r³+9r²+25r+17 = 0

Using synthetic division, we get:

(r+1)(r²+8r+17) = 0

The roots are:

r1 = -1, r2 = -4+3i, r3 = -4-3i

Hence, the homogeneous solution is:

y_h(t) = c1e^(-t) + c2e^(-4t)cos(3t) + c3e^(-4t)sin(3t)

To find the particular solution, we assume a solution of the form:

y_p(t) = A cos(3t) + B sin(3t)

Taking the derivatives and substituting in the differential equation, we get:

-8A/3 - 8B = 0
-25A/9 + 25B/3 = 5/9

Solving the system of equations, we get:

A = 5/18, B = -5/54

Hence, the particular solution is:

y_p(t) = (5/18)cos(3t) - (5/54)sin(3t)

The total solution is the sum of the homogeneous and particular solutions:

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

Substituting the initial conditions, we get:

c1 = 1, c2 = 0, c3 = 1/3

Hence, the total solution is:

y(t) = e^(-t) + (1/3)e^(-4t)sin(3t) + (5/18)cos(3t) - (5/54)sin(3t)

 

[LenXine]

pa mark as solution and upvote po thankieeee

To solve the given differential equation using the Laplace Transform and Classical Method, we'll follow these steps:

Given differential equation: D³y(t) + 9D²y(t) + 25Dy(t) + 17y(t) = 5cos(3t)
Initial conditions: D²y(0) = 0, Dy(0) = 1, y(0) = 0

1. Laplace Transform Method:

Take the Laplace transform of both sides of the equation. The Laplace transform of the derivatives becomes:

L{D³y(t)} = s^3Y(s) - s^2y(0) - sy'(0) - y''(0) = s^3Y(s) - s

L{D²y(t)} = s^2Y(s) - sy(0) - y'(0) = s^2Y(s) - 1

L{Dy(t)} = sY(s) - y(0) = sY(s)

L{y(t)} = Y(s)

L{cos(3t)} = s/(s^2 + 9)

Now, apply the Laplace transform to the given differential equation:

s^3Y(s) - s + 9(s^2Y(s) - 1) + 25(sY(s)) + 17Y(s) = 5(s/(s^2 + 9))

Solve for Y(s):

Y(s) [s^3 + 9s^2 + 25s + 17] = 5s/(s^2 + 9) + s + 9

Now find the inverse Laplace transform of Y(s) to get the solution y(t).

2. Classical Method:

For this method, we'll first find the complementary function and then the particular solution. We'll combine both to find the total solution. We have the homogeneous equation:

D³y(t) + 9D²y(t) + 25Dy(t) + 17y(t) = 0

The auxiliary equation is: r^3 + 9r^2 + 25r + 17 = 0

Solve for r to find the complementary function y_c(t).

For the particular solution, let's assume y_p(t) in the form:

y_p(t) = A*cos(3t) + B*sin(3t)

Differentiate y_p(t) w.r.t t to find y'_p(t) and then again differentiate y'_p(t) w.r.t t to find y''_p(t). Differentiate y''_p(t) w.r.t t to find y'''_p(t). Substitute y_p(t), y'_p(t), y''_p(t), and y'''_p(t) into the given differential equation and solve for A and B.

Finally, the total solution will be:

y(t) = y_c(t) + y_p(t)

Both the methods give the desired solutions. However, without specific information about roots or the nature of the solution in the classical methods, it would be difficult to provide a complete analytical solution.