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Find the general solution of the following nonhomogeneous second order differential equation: y" - 4y = e^2x

User Kristofer
by
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1 Answer

5 votes

Answer:

Solution is
y=c_(1)e^(2x)+c_(2)e^(-2x)+(1)/(4)xe^(2x)

Explanation:

the given equation y''-4y
=e^(2x) can be written as


D^(2)y-4y=e^(2x)\\\\(D^(2)-4)y=e^(2x)\\\\

The Complementary function thus becomes

y=c_{1}e^{m_{1}x}+c_{2}e^{m_{2}x}

where
m_(1) , m_(2) are the roots of the
D^(2)-4

The roots of
D^(2)-4 are +2,-2 Thus the comlementary function becomes


y=c_(1)e^(2x)+c_(2)e^(-2x)

here
c_(1),c_(2) are arbitary constants

Now the Particular Integral becomes using standard formula


y=(e^(ax))/(f(D))\\\\y=(e^(ax))/(f(a)) (f(a)\\eq 0)\\\\y=x(e^(ax))/(f'(a))(f(a)=0)


y=(e^(2x))/(D^(2)-4)\\\\y=(e^(2x))/((D+2)(D-2))\\\\y=(1)/(D-2)* (e^(2x))/(2+2)\\\\y=(1)/(4)* (e^(2x))/(D-2)\\\\y=(1)/(4)xe^(2x)

Hence the solution is = Complementary function + Particular Integral

Thus Solution becomes
y=c_(1)e^(2x)+c_(2)e^(-2x)+(1)/(4)xe^(2x)

User ChuckSaldana
by
8.3k points
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