What is a solution to the differential equation dy/dx=x+y?

1 Answer
Oct 19, 2016

y = C_1e^x-x-1

Explanation:

Let u = x + y

=> (du)/dx = d/dx(x+y) = 1+dy/dx

=> dy/dx = (du)/dx-1

Thus, making the substitutions into our original equation,

(du)/dx-1 = u

=> (du)/(u+1) = dx

=> int(du)/(u+1)=intdx

=> ln(u+1) = x + C_0

=> e^(ln(u+1)) = e^(x+C_0)

=> u+1 = C_1e^x" " (where C_1 = e^(C_0))

Substituting x+y = u back in,

=> x + y + 1 = C_1e^x

:. y = C_1e^x-x-1