Thread: Differentiate Solution and Match Up with Equation

1. Differentiate Solution and Match Up with Equation

For C, can someone show me step-by-step how to isolate the function y and/or its derivatives? I said "and/or" just in case, I cannot isolate the function y itself but can for its derivatives so that I can say that the equation C is the general solution of one the numbers.

Any help would be greatly appreciated!

2. Re: Differentiate Solution and Match Up with Equation

Use implicit differentiation and the product rule. What do you get as a first step?

3. Re: Differentiate Solution and Match Up with Equation

Originally Posted by s3a
For C, can someone show me step-by-step how to isolate the function y and/or its derivatives? I said "and/or" just in case, I cannot isolate the function y itself but can for its derivatives so that I can say that the equation C is the general solution of one the numbers.

Any help would be greatly appreciated!
1. is a standard second order linear constant coefficient ODE. How do you solve those?

2. Rewrite it as

\displaystyle \begin{align*} \frac{dy}{dx} &= -\frac{2xy}{x^2 - 6y^2} \\ \frac{dx}{dy} &= \frac{6y^2 - x^2}{2xy} \\ \frac{dx}{dy} &= \frac{6y^2}{2xy} - \frac{x^2}{2xy} \\ \frac{dx}{dy} &= 3\left(\frac{y}{x}\right) - \frac{1}{2}\left(\frac{x}{y}\right) \end{align*}

and then make the substitution $\displaystyle v = \frac{x}{y}$...

3. Another second order linear constant coefficient ODE...

4. The DE is separable.

5. The answer will be the whichever one is left over...

4. Re: Differentiate Solution and Match Up with Equation

I'm attaching the pdf which contains that segment of my work but I am now stuck at the next step. I also don't see how it's useful in differential equations to have solutions with y not isolated even though the derivatives y' (and so on) are isolated or basically if at least one of the equations has a problem with the isolation.

5. Re: Differentiate Solution and Match Up with Equation

So, divide top and bottom of the RHS of your final result by -3. Does that start to look more familiar?

I also don't see how it's useful in differential equations to have solutions with y not isolated even though the derivatives y' (and so on) are isolated or basically if at least one of the equations has a problem with the isolation.
You can still obtain useful information about the solutions, even if they're defined implicitly. For example, computers can graph implicit solutions.

6. Re: Differentiate Solution and Match Up with Equation

Ackbeet: I noticed that C matches #2 when I divide top and bottom by -3. Does this mean that the antiderivative of #2 is the general solution for differential equation #2?

Prove It: You mentioned a few stuff that I did not cover so I'll look at it when I learn the new stuff to get this question from a different perspective so thank you still.

7. Re: Differentiate Solution and Match Up with Equation

Originally Posted by s3a
Ackbeet: I noticed that C matches #2 when I divide top and bottom by -3. Does this mean that the antiderivative of #2 is the general solution for differential equation #2?
I wouldn't quite phrase it that way. You can't integrate # 2 directly. Say, rather, that Equation C is the general solution of DE # 2. On the other hand, the process of solving differential equations is essentially integrating. So maybe you can say that the integral of DE # 2 is Equation C.