Should the derivative of a function correspond to its differential equation?

Question

I'm starting to learn about differential equations through some Khan Academy videos. In one of the first he gives the following differential equation $dy/dx = -2x + 3y -5$ and finds a solution as the function $y= (2/3)x + 17/9$. Now if I take the derivative of the latter it doesn't give me the original differential equation which I thought it would. Why is this the case?

I should say I thought this would be so because I've seen cases where the way to get the solution for the DE is to integrate it. Thus, I assumed by reversing the process - differentiating the equation - I'd arrive back at the DE.

Answer 1

The only guarantee is that if you substitute a (correct) solution for a differential equation back into that equation, you will find that the equation is satisfied.

This is the case here.

The solution is $y = \frac 23 x + \frac{17}9$

The right hand side of the differential equation is $-2x + 3y - 5 = -2x + 3(\frac 23 x + \frac{17}9) - 5 = \frac 23$.

The left hand side of the differential equation is $\frac{dy}{dx} = \frac 23$.

Since right and left hand sides work out to be the same, the differential equation is satisfied by the putative solution, so the solution is obviously correct.

Answer 2

Clearly, solution $y=\frac{2}{3}x + \frac{17}{9} $ is satisfying your ODE. May be you are forgetting to substutute y on right hand side.