Question

Why do we need two solutions to solve a 2nd order linear differential equation?

Answer 1

@irishboy123

Answer 2

my input is:

Solving this equation essentially involves integrating twice.

Each integration involves a 'constant of integration' so you need to known points to calculate those constants.

(Any comments welcome - this is not a 'current topic' for me - I may have forgottent the key points...)

Answer 3

I am not talking about constants. I am talking about solution
Example lets consider a differential equation with equal roots for auxiliary equation.
So the reasoning behind why cant we use \[y=Ae^{n_1x}+ Be^{n_2x}\]as its general solution is because since the roots are equal we get a single solution \[y=Ce^{nx}\]. So my question is what's the problem with having one solution. After all it's A SOLUTION.

Answer 4

Let me ask you a relatively simple and familiar question.

How many coordinate axes do you need to locate all the points on the xy plane ?

Answer 5

2

Answer 6

Yes. How many would you be needing if you wanted to locate all the points in 3D ?

Answer 7

3

Answer 8

Could you tell me why/how you think your answers are correct ?

Answer 9

Well because in an xy plane, the solution can be made up of both x and y components. The same reasoning can be applied for xyz plane

Answer 10

Nice, let me ask you another question. Again simpler one, but it takes us in the right direction.

Answer 11

Can you find the solutions of below ?
x + y + z = 1

Answer 12

There are infinite solutions

Answer 13

Yes, but that is a bit vague. I want to know more about the kind of solutions that equation has.

Answer 14

What geometrical shape does that equation represent ?

Answer 15

Well the solution can have imaginary and real roots.

Answer 16

A line in a 3D plane

Answer 17

Assume we're working with reals for now

Answer 18

Not really. Hasn't your textbook talked about planes yet ?

Answer 19

If not, its okay. We can discard this idea and think of somehting else :)

Answer 20

Oh I am sorry, a plane in a 3D box

Answer 21

Good. x + y + z = 1 represents a 2D plane.

Imagine you're walking on this plane. You would be needing a minimum of 2 different axes in order to reach every point on the plane, right ?

Answer 22

I assume we can change the direction of axes in the above example? Like if the plane is tilted, we can tilt our axes. Otherwise for some we would be needing 3

Answer 23

3 or 4 or 5 also work. But 2 axes will do. Just pick these axes on the plane itself !

Answer 24

Yes. I was suggesting the same

Answer 25

Lets say the unit vectors along those two axes are e1 and e2.

Answer 26

Okay