Who is asking: Student
Level: All
Question:
I have a question that i really cant do, it is as follows:
The ODE dy/dx + 0.5y = 0.5e-1.5x ; y(5) = 2
Solve the ODE subject to the given condition using exact methods and evaluate the solution y for x = 5 x=5.2 x=5.4 x=5.6 x=5.8 x=6
I think i should use an integrating factor but i cant do it. I hope someone can help.
Hi David,
First you need to solve the homogenous equation
dy/dx + 0.5y = 0
You can solve this equation using the integration factor
e-integral(0.5)dx = e-0.5x
I actually looked at the equation as
dy/dx = -0.5y
and realized that y is a function which is almost identical to its own derivative so I substituted
y = eax
into the equation and found that
a = -0.5
In either case you get the general solution of the homogenous equation to be
y = k e-0.5x
Now you need one solution of
dy/dx + 0.5y = 0.5e-1.5x
Since the right side is 0.5e-1.5x I guessed a solution of the form
y = A e-1.5x
Substitute this into dy/dx + 0.5y = 0.5e-1.5x and solve for A. When I did this I got A = - 0.5. Hence the general solution of
dy/dx + 0.5y = 0.5e-1.5x
is
y = k e-0.5x - 0.5 e-1.5x
Harley