This is exercise 18 of Section 23 page 55 Solve the linear i

This is exercise 18 of Section 2.3 page 55. Solve the linear initial value problem. Dy/dx + 4y - e^-x = 0, y(0) = 4/3 y = ___________

dy/dx we denote here as y\'

SO we ahve

y\'+4y=e^{-x}

Integrating factor is e^{4x}

Multiplying by the integrating factor gives

(y\'+4y)exp(4x)=exp(3x)

(yexp(4x))\'=exp(3x)

Integrating gives

y exp(4x)=exp(3x)/3+C

y=exp(-x)/3+C exp(-4x)

y(0)=4/3=1/3+C

Hence, C=1