The solution to the initial value problem dy/dx - y = e^3x with y(0) = 3 involves using the integrating factor method. The integrating factor is I(x) = e^∫-1 = e^-x. The correct manipulation leads to the equation e^-x(dy/dx) - e^-x . y = e^2x, which integrates to yield y = C + 2e^3x. However, the proposed solution does not satisfy the initial condition, indicating a mistake in the calculations.
PREREQUISITESStudents studying differential equations, mathematics educators, and anyone looking to strengthen their problem-solving skills in calculus and differential equations.
Yes, in this case p(x)=-1. However, you're answer isn't correct as it doesn't satisfy the initial value problem. You have made a little slit in the second to last line of your working.mad_monkey_j said:Homework Statement
Find the solution to the initial value problem
dy/dx - y = e^3x
y(0) = 3
Homework Equations
e^∫p(x)
The Attempt at a Solution
Do I treat p(x) = -1?
I(x) = e^∫-1 = e^-x
e^-x(dy/dx) - ye^-x = e^3x . e^-x
e^-x(dy/dx) - e^-x . y = e^2x
e^-x . y = ∫e^2x
y = (2e^2x + c)/(e^-x)
y = C+2e^3x?