Find the general solution of the differential equation and g

Find the general solution of the differential equation and give the largest interval over which the general solution is defined

(x+4)dy/dx + (x+5)y = 5xe^-2x

Solution

Ans;

You have:

(x+4)(dy/dx) + (x+5)*y = 5x*exp(-2x)

(dy/dx) + ((x+5)/(x+4))y = 5x*exp(-2x)/(x+4)

Expand (x+5)/(x+4) in terms of partial fractions:

(dy/dx) + (1+ 1/(x+4))y = 5x*exp(-2x)/(x+4)

This is a first-order, linear ODE that we can try to solve using an integrating factor. The integrating factor, p(x) is given by:

p(x) = exp(INTEGRAL of {(1+ 1/(x+4)) dx}

p(x) = exp(x + ln(x+4))

p(x) = (x+4)*exp(x)

The solution is then given by:

y(x) = (exp(-x)/(x+4))*INTEGRAL of {(x+4)*exp(x) * 5x*exp(-2x)/(x+4) dx}

y(x) = (exp(-x)/(x+4))*INTEGRAL of {5x*exp(-x) dx}

y(x) = (exp(-x)/(x+4))*[c - 5*(x+1)*exp(-x)]

y(x) = [c*exp(-x) - 5x*exp(-2x) - 5*exp(-2x)]/(x+4)