Eliminate the arbitrary constants

Question

Please do leave a solution so I can study it, thank you!

1. Cxsiny+x²y=Cy
2. y=Ae^-3x-Be^2x+Cx³
3. Ay=e^{Bx^2}

Answer 1

Hi Azalea,

From the comments it seems the objective is to eliminate the constants by differentiating these equations to arrive at new equations? I can try to start you off. The general way I'd begin my approach is by noticing there are a lot of exponential functions, and this means we can exploit the fact that d/dx of e^x is e^x. Furthermore, we will use the fact that the derivative of a constant is zero.

Let me show the solution to (3) for example.

(a.) from equation (3) if we differentiate implicitly we get Ay' = 2Bx exp(Bx^2). Noticing we can now substitute the exponential ( (3) Ay = exp(Bx^2) to arrive at the equation Ay' = 2Bx (Ay). Constant A can be divided away from both sides, and now we have an explicit solution for the constant,

B = (y')/(2xy).

(b.) So we killed the constant A, and now we have isolated B on the left-hand side of the equation. If we differentiate once more, B will be gone. Just remember we have to use implicit differentiation, and this step is easy to make a mistake, because can use a combination of product and quotient rules. I just used product rule like this, d/dx{ 1/(2xy) } y' + 1/(2xy) d/dx{ y' }. We arrive at

0 = - { (2y+2xy') y' }/{ (2xy)^2 } + { y'' }/{ 2xy }

so a common factor to all terms is 1/(2xy) we can get rid of it and simplify a bit to arrive a the following nonlinear second order differential equation

xyy'' = yy' + x(y')^2

or if you prefer

y'' = y'/x + (y')^2/y

Try to verify my answer and then apply these common themes to the other problems, let me know what you find.