Differential Equations Find a particular solution to the lin

Differential Equations

Find a particular solution to the linear system using the given initial values.

x\' = y ; x(0) = 1

x\' + y\' = -x + y ; y(0) = 1

Solution

Solution: 29/04/2016.

X¹ = y

dx/dt = y

Dx –y = 0 --------------------------- (1)

X¹ + y¹ =-x+y

dx/dt + x + dy/dt –y = 0

( D+1) x + ( D-1) y = 0 --------------------(2)

Multiply equation (1) with D+1, multiply equation (2) with D

   (1)x (D+1) – (2) x D

solving (1) and (2)

D (D+1 ) x – (D+1) y = 0 --------------------(3)

D (D+1) x + D ( D-1) y =0 ---------------------(4)

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F(D) = D² + 1

The A.E is f(m) = m² +1

M= +I, -i

Y_c = C₁ cos X + C₂ sin x

Solution of ( 6) is y_c = C₁ cos X + C₂ sin x

Now   I       D         -1         I = 0

            I      D+1   D-1        I

Given   x((0) = 1 and y(0) = 1

1= C₁ cos (0) + C₂ sin (0 )

1 = C₁

Therefore G.S of (6) is 1= Cos x +C.