I'm starting to learn about differential equation

[cambo86]

Homework Statement

Verify that the differential equation,
${\frac{dy}{dx}} = 15 - {\frac{2y}{81+3x}}$
has the general solution
$y(x) = 3(81+3x) + C(81+3x)^{-2/3}$

2. The attempt at a solution
I've just learned about differential equations, so I'm probably missing something very basic. I've tried serperating x and y so that I can integrate and it's not an exact equation.

Thanks in advance for the help.

 

[leright]

Homework Statement

Verify that the differential equation,
${\frac{dy}{dx}} = 15 - {\frac{2y}{81+3x}}$
has the general solution
$y(x) = 3(81+3x) + C(81+3x)^{-2/3}$

2. The attempt at a solution
I've just learned about differential equations, so I'm probably missing something very basic. I've tried serperating x and y so that I can integrate and it's not an exact equation.

Thanks in advance for the help.

Can you differentiate the solution and plug it back into the original differential equation and see if it satisfies the DE?

-Matt

 

[CAF123]

If you want to solve the differential equation, bring the fractional term on the RHS over to the LHS and you see you have a form that can be solved via integrating factors (the resulting eqn is linear in the independent variable y so this method is valid).

 

[tiny-tim]

hi cambo86!

Verify …

"verify" always means that you don't have to solve it, you just assume the answer, and check it!

(by plugging it in, as Matt (leright) says)

 

[cambo86]

So I differentiated the general solution and then subsituted that into the DE. Then I manipulated that to form the general solution again. Is that all that is needed to verify?

 

[Ray Vickson]

So I differentiated the general solution and then subsituted that into the DE. Then I manipulated that to form the general solution again. Is that all that is needed to verify?

I don't understand what you are trying to say. You have a function
$$y = y(x) = 3(81+3x)+C(81+3x)^{−2/3}.$$
You can compute dy/dx. When you do that, can you re-write dy/dx as
$$15− \frac{2y}{81+3x}?$$
If your answer is YES, then you have verified the solution. What other possible meaning could the word "verify" have?