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# Solving Second Order Linear Differential Equation by Using D-operator

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Use D-operators to find a particular solution to the differential equation:
y^n + y' - 2y= e^-2x

Hence write its general solution. Find the solution that satisfies the initial conditions:
y(0) = 1/3, y'(0) = -1/3

https://brainmass.com/math/ordinary-differential-equations/solving-second-order-linear-differential-equation-d-operator-576598

## SOLUTION This solution is FREE courtesy of BrainMass!

Solution:
The characteristic equation is

so the complementary solution will be

Given

Let f(D) =
Since f(-2) = 0
f'(D) = 2D+1
f'(-2) = 2(-2)+1 = -3

So particular integral will be

so the general solution will be

Given y(0) = 1/3

Now,
Given y'(0) = -1/3

Subtract equation (ii) from equation (i)
3c1 = 1/3
c1 = 1/9

Thus, general solution will be