Question

7.
= (x - 1)?. If y= 0 when x = 0.
dx​

Answer 1

They are "First Order" when there is only dydx , not d2ydx2 or d3ydx3 etc

Linear

A first order differential equation is linear when it can be made to look like this:

dydx + P(x)y = Q(x)

Where P(x) and Q(x) are functions of x.

To solve it there is a special method:

We invent two new functions of x, call them u and v, and say that y=uv.

We then solve to find u, and then find v, and tidy up and we are done!

And we also use the derivative of y=uv (see Derivative Rules (Product Rule) ):

dydx = u dvdx + v dudx

Steps

Here is a step-by-step method for solving them:

1. Substitute y = uv, and

dydx = u dvdx + v dudx

into

dydx + P(x)y = Q(x)

2. Factor the parts involving v

3. Put the v term equal to zero (this gives a differential equation in u and x which can be solved in the next step)

4. Solve using separation of variables to find u

5. Substitute u back into the equation we got at step 2

6. Solve that to find v

7. Finally, substitute u and v into y = uv to get our solution!