Checking Your Answers

One of the great things about Diff EQs is you can always check your answer. The checks aren't necessarily super easy, though, so I'd recommend waiting until the end of an exam and checking if you still have time left over. Your professor will probably take off points for algebraic or differentiation errors, so checking is definitely a good idea!

Say you're solving the equation:

y'' - 10y' + 25y = 0; y(0) = 1, y'(0) = 0

And you've come up with the following solution:

y(x) = e^5x - 5xe^5x

First make sure your first initial condition holds: y(0) = 1

y(0) = e^5*0 - 5*0*e^5*0y(0) = 1 - 0 = 1

Now take the derivative of your equation:

y'(x) = 5e^5x - 5e^5x - 25xe^5x     (notice that you must use product rule on the second term)
y'(x) = -25xe^5x

Check your second initial condition: y'(0) = 0

y'(0) = -25*0*e^5*0y'(0) = 0

Now take the second derivative:

y''(x) = -25e^5x - 125xe^5x      (product rule again)

Now you have y'', y', and y - you can substitute these into the original equation and make sure it all equals zero.

y'' - 10y' + 25y = 0
-25e^5x - 125xe^5x - 10(-25xe^5x) + 25(e^5x - 5xe^5x) = 0
-25e^5x - 125xe^5x + 250xe^5x + 25e^5x - 125xe^5x = 0

Notice the 1st and 4th terms cancel, and the 2nd, 3rd, and 5th terms cancel, so the whole thing equals 0, and your solution checks out. Go ahead and try this anytime you're doubting yourself!

Happy calculating,
Rachel