Question

Form a differential equation if y=Axe^x

Answer 1

We can differentiate this to form a differential equation. We get:

(assuming A is a constant):

y' = Ae^x + Axe^x. [ Do not forget the product rule, buddy]

Now note that when you solve this ODE you get a family of solutions i.e, y=Axe^x + C which contains your y for the case C=0; The only way I know to avoid it is also give an initial condition and make it an IVP problem for the sake of completeness. Since this is a 1st order linear ODE, it has one parameter of solutions, and it needs one initial initial condition y(x0) = y0

I'd say y(1) = Ae is a good initial condition.

Hope this helps :)

Answer 2

Since this is the multiplication of two variables, you must use the product rule, which states if y = u*b, in which u and b are two variables, then y' = u*b' + u'*b.

Therefore, y' = A(x*eˣ + x'*eˣ). Since the derivative of eˣ is eˣ, that term appears unchanged in both terms. However, since x' is another way of writing dx/dx, x' equals 1, so it is unnecessary to write, leaving you y' = A(xeˣ + eˣ).

You may leave your answer written like that, or you can choose to distribute the constant A, giving you y' = Axeˣ +Aeˣ.

Hope this helps!