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C. Convolution Method

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The convolution of two functions g andfis the function g "f defined by

(g * f)(t) = J>(t - v)f(v)dv .

The aim of this project is to show how convolutions can be used to obtain a particular solution to

a nonhomogeneous equation of the form

(11) ay" + by' + cy = f(t) ,

where a. b. and c are constants, a i=O.

(a) Use Leibniz's rule

d/dt integral from a to t: h(t, v)dv = a to t dh/dt: (t, v)dv + h(t, t) ,

to show that

(Y*f)'(t) = (i *f)(t) -;-y(O)f(t)

and

(y' f)"(t) = (y" *f)(t) -+-y'(O)f(t) + y(O)f'(t) ,

assuming y andf are sufficiently differentiable.

(b) Let y,(t) be the solution to the homogeneous equation ay" + bv' + cv = 0 that satisfies

y,(O) = 0, y~(O) = l/a. Show that Ys* f is the particular solution to equation (II)

satisfyingy(O) = i(O) = o.

(c) Let Yk(t) be the solution to the homogeneous equation av" + by' + C\ = 0 that satisfiesy(

O) = Yo,y'(O) = Y1, and letysbe as defined in part (b). Show that

(Ys*f)(t) +

(12)

is the unique solution to the initial value problem

ay" + by' + cv = f(t); y(O) = Yo ' i(O) = Y1 .

(d) Use the result of part (c) to detennine the solution to each of the following initial value

problems. Carry out all integrations and express your answers in terms of elementary

functions.

(i) + Y = tan t ; y(O) = 0 . )"(0) = -I

(ii) 2y" -+-y' - Y = e -, sin t ; v( 0) = I, y' (0) = I .

(iii) -2y'+y=vte'; v(0)=2, y'(O)=O

i'

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# Convolution Method

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