First and Second Order Differential Equation

[aznkid310]

Homework Statement

a) Show that phi(t) = e^2t is a solution of y' - 2y = 0 and that C*phi(t) is also a solution for any constant C

b) Show that phi(t) = 1/t is a solution of y' + y^2 = 0 for t>0 but that y = c*phi(t) is not a solution unless c = 0 or c = 1

Homework Equations

What do they mean by "show'?

The Attempt at a Solution

a) dy/2y = dt

(1/2)ln(y) = t + C

y = e^(2t + c) = Ce^2t, where c is any constant

Now what?

b) -dy/y^2 = dt

1/y = t + C

y = 1/(t + C)

Again, how do i show?

 

[tiny-tim]

Hi aznkid310!

a) dy/2y = dt

(1/2)ln(y) = t + C

y = e^(2t + c) = Ce^2t, where c is any constant

Now what?

That's it! You've done it!

b) -dy/y^2 = dt

1/y = t + C

y = 1/(t + C)

Again, how do i show?

Again, that's it … except of course you still have to make the obvious remark that C/t is not a solution!

What do they mean by "show'?

I think they just mean "prove".

Alternatively, they may mean "no need to pretend you don't know the answer … just plug the answer straight into the original equation and confirm that it works"

Better ask your teacher/tutor/professor which one it is!

 

[Dick]

'Show' just means substitute the given solutions into the differential equations and verify that they work or not. You don't have to solve the differential equation.