Find all solutions to the following differential equations d

Find all solutions to the following differential equations:

(d) 3y\'+y=2e^-x

(e) y\'+3y=e^ix

Solution

d

We can rewrite as

y\'+y/3=2/3 e^{-x}

Integrating factor is

exp(x/3)

Multiplying by this integrating factor gives

(y\'+y)exp(x/3)=2/3 exp(-2x/3)

(y exp(x/3))\'=2/3 exp(-2x/3)

INtegrating gives

y exp(x/3)=-exp(-2x/3)+C

y=-exp(-2x/3)+Cexp(-x/3)

e)

Integrating factor is exp(3x)

Multiplying by this integrating factor gives

(y\'+3y)exp(3x)=exp((3+i)x)

(y exp(3x))\'=exp((3+i)x)

Integrating gives

y exp(3x)=exp((3+i)x)/(3+i)+C

y=exp(ix)/(3+i)+C exp(-3x)