SOLUTION: Please help me answer the following question: (a) Determine the particular solution of the equation {{{ d^2y/dx^2 - 3(dy/dx) = 9 }}} given the initial conditions: y(0) =

Algebra ->  Customizable Word Problem Solvers  -> Evaluation -> SOLUTION: Please help me answer the following question: (a) Determine the particular solution of the equation {{{ d^2y/dx^2 - 3(dy/dx) = 9 }}} given the initial conditions: y(0) =       Log On

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Question 1115567: Please help me answer the following question:
(a) Determine the particular solution of the equation
+d%5E2y%2Fdx%5E2+-+3%28dy%2Fdx%29+=+9+
given the initial conditions:
y(0) = 0, y'(0) = 0
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Using Laplace transforms:

L(y'') = +s%5E2+Y(s) - sy(0) - y'(0)
L(y') = sY(s) - y(0)
L(c) = c/s (c=a constant)

Noting y'(0)=y(0)=0, the Laplace transform is:
++s%5E2Y%28s%29+-+3sY%28s%29+=+9%2Fs+

Now we "just" need to isolate Y(s) and then take the inverse Laplace transform.

+Y%28s%29+=++%289%2F%28s%2A%28s%5E2-3s%29%29%29+
++Y%28s%29+=++%289%2F%28s%5E2%2A%28s-3%29%29%29+

Use partial fraction expansion to get the right hand side into a form in which the inverse Laplace can be taken:
++%289%2F%28s%5E2%2A%28s-3%29%29%29+=+A%2Fs+%2B+B%2Fs%5E2+%2B+C%2F%28s-3%29+ (1)
Multiply both sides by +s%5E2%28s-3%29+ :
++9+=++A%2A%28s%2A%28s-3%29%29+%2B+B%2A%28s-3%29+%2B+C%2As%5E2+
++9+=++%28As%5E2+-+3As%29+%2B+%28Bs+-+3B%29+%2B+Cs%5E2+

From this you get three equations in three unknowns:
A+C = 0 (from the s%5E2 terms)
-3A+B = 0 (from the s%5E1 terms)
-3B = 9 (from the s%5E0 terms)
—> B = -3 —> A= -1 —> C = 1


So we can write (1) as:
++Y%28s%29+=+-1%2Fs+%2B+-3%2Fs%5E2+%2B+1%2F%28s-3%29++
Taking the inverse Laplace gives:
+highlight%28++y%28x%29++=++-1+-3x+%2B+e%5E%283x%29+%29+
——
Check:
Initial conditions:
y(0) = -1-3*0+e^(0) = -1 + 1 = 0 (ok)
y'(x) = -3 + 3e^(3x) and y'(0) = -3 + 3e^(0) = -3+3 = 0 (also ok)
Entire equation:
y''(x) = 9e^(3x)
y'' - 3y' = 9e^(3x) - 3(-3+3e^(3x)) = 9e^(3x) + 9 -9e^(3x) = 9 (ok)
———————————
My answer is the general solution. A particular solution is often guessed at the start, and then combined with the homogeneous solution (i.e. particular solution would be a function y(x) that satisfies y''-3y' = 9 while the homogenous ("complementary") solution would satisfy y''-3y' = 0 and you add the two solutions together to get the general solution. I don't know how to guess a proper particular solution for this problem. One could guess y(x) = Ae^(kx) + Bxe^(mx) + C, I suppose, but I wouldn't know to guess that without seeing the general solution first.