1. Solve the given differential equation by undetermined coefficients.
y'' − 8y' + 16y = 12x + 6
y(x) =_____
2. Solve the given boundary-value problem.
y'' + 7y = 7x, y(0) = 0, y(1) + y'(1) = 0
y(x) = _____
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Expert's answer
2020-07-05T17:33:17-0400
1.The given equation can be written as (D2−8D+16)y=12x+6.
The auxiliary equation is, m2−8m+16=0. Solving we get, m=4(twice) . Thus, the complementary function is, yc=(c1x+c2)e4x. Let yp=Ax+B be the trial particular solution. Then,
Thus, the general solution is, y=yc+yp=(c1x+c2)e4x+43(x+1).
2.The given equation can be written as (D2+7)y=7x. The auxiliary equation is, m2+7=0. Solving we get, m=±i7. Thus, the complementary function is, yc=c1cos(7x)+c2sin(7x). The particular solution is
yp=D2+717x=1+7D21x=(1+7D2)−1x=x
y=c1cos(7x)+c2sin(7x)+x is the general solution.
Given the boundary conditions, y(0)=0;y(1)+y′(1)=0. Using these conditions we get,
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