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Question 1115049: Please help me answer the following question:
(a) Determine the particular solution of the equation
given the initial conditions:
y(0) = 0, y'(0) = 0
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! this is a second-order linear differential equation which is nonhomogeneous
:
Note the general form is
:
1) a(d^2 y/dx^2) + b(dy/dx) + c(x)y = G(x) where a, b, c, are constants and G is a continuous functions not equal to 0 for some x
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The complementary form is
:
2) a(d^2 y/dx^2) + b(dy/dx) + c(x)y = G(x) where a, b, c, are constants and G is a continuous functions equal to 0 for some x
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The general solution is given by the formula
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y(x) = y(p) (x) + y(c) (x), where y(p) is a particular solution to equation 1 and y(c) is a general solution to equation 2
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we are given the problem
:
(d^2 y/d x^2) - 3(dy/dx) = 9
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solve the equation 2 form
r^2 -3r = 0
:
r(r-3) = 0
:
r=0 and r=3
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So the solution to equation 2 is
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y(c) = c(1)e^(0 * x) + c(2)e^-3x = c(1) +c(2)e^(-3x)
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since G(x) = 9 is a linear equation, we are looking for a particular solution of the form
:
y(p) (x) = Bx +C
:
y'(p) = B and y''(p) = 0
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now substitute into the given equation
:
0 -3B = 9
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polynomials are equal when their coefficients are equal
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B = -3
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the general solution is
:
y(x) = c(1) + c(2)e^(-3x) + 9
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y(0)=0, then
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c(1) +c(2) +9 = 0
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y' = -3c(2)e^(-3x)
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y'(0) = 0
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0 = -3c(2)
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c(2) = 0 and c(1) = -9
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the solution to the initial value problem is
:
y(x) = -9 +0 +9 = 0
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