Solve the initial value problem: dy/dt-y=7e^t + 10e^6t with y(0)=4
y = ?
you know that the solution to dy/dt - y = 0 is
y = c e^t
that means that the solution to y'-y=7e^t + 10e^6t is
y = c e^t + 7t e^t + 2e^6t
To solve the initial value problem dy/dt - y = 7e^t + 10e^6t with y(0) = 4, we can follow these steps:
Step 1: Find the integrating factor.
The integrating factor is given by the exponential of the integral of the coefficient of y, which in this case is -1.
∫(-1)dt = -t
Therefore, the integrating factor is e^(-t).
Step 2: Multiply both sides of the differential equation by the integrating factor.
e^(-t) * (dy/dt - y) = e^(-t) * (7e^t + 10e^(6t))
This simplifies to:
e^(-t) * dy/dt - e^(-t) * y = 7 + 10e^(5t)
Step 3: Apply the product rule on the left side of the equation.
d(e^(-t) * y)/dt = 7 + 10e^(5t)
Step 4: Integrate both sides of the equation with respect to t.
∫d(e^(-t) * y)/dt dt = ∫(7 + 10e^(5t)) dt
This simplifies to:
e^(-t) * y = 7t + (2/5)e^(5t) + C
Step 5: Solve for y.
Multiply both sides by e^t to isolate y:
y = (7t + (2/5)e^(5t) + C) * e^t
Step 6: Apply the initial condition y(0) = 4.
Plug in t = 0 and y = 4 into the equation and solve for C:
4 = (7(0) + (2/5)e^(5(0)) + C) * e^(0)
4 = (2/5) + C
Simplifying:
C = 4 - (2/5)
C = 18/5
Step 7: Substitute the value of C back into the equation.
y = (7t + (2/5)e^(5t) + 18/5) * e^t
Therefore, the solution to the initial value problem is:
y = (7t + (2/5)e^(5t) + 18/5) * e^t