Question

Using the D-operator Method, get complete specific solutions, including the constants of integration, for the following equation:

3 dx/dt = 5x + 7 x(0+) = -2

Answer 1

Let d/dt=T

So equation is

3T(x)=5x+7

Applying the operator again to above equation

3T(T(x))=5T(x)+T(7)

3T^2(x)=5T(x)+0

(3T^2-5T)(x)=0

(3T-5)T(x)=0

So this is the factorisation of the operator

(3T-5)(x)=C

3Tx-5x=C

x'-(5/3)x=C/3

INtegrating factor is exp(5t/3)

Multiplyinbg gives

(x'-5x/3) exp(5t/3)=C exp(5t/3)/3

(x exp(5t/3))'=C exp(5t/3)/3

Integrating gives

x exp(5t/3)=C exp(5t/3)/5+D

x=C/5+D exp(-5t/3)

x(0)=C/5+D=-2

dx/dt(0)=5x(0)+7=5*(-2)+7=-3

dx/dt=-5D exp(-5t/3)/3

dx/dt(0)=-3=-5D/3

D=9/5

C/5+D=-2

C/5=-2-9/5=-19/5

C=-19

x=-19/5+9/5 exp(-5t/3)