Modeling and verifying differential equations

Understand the idea

A differential equation relates a quantity to one or more derivatives. Translate proportional growth as a derivative proportional to the quantity; a rate depending only on time is a different model. To check a proposed solution, compute its derivative and substitute both function and derivative into the equation throughout the claimed domain. An initial condition picks a member of a solution family when uniqueness holds.

\frac{dy}{dt}=ky\quad\text{models a rate proportional to }y

Worked example

Verify that $\displaystyle y=Ce^{3t}$ solves $\displaystyle y'=3y$ .

Differentiate:

\displaystyle y'=3Ce^{3t}

. The right side is

\displaystyle 3y=3Ce^{3t}

, so the equation holds for every

\displaystyle t

and every real constant

\displaystyle C

. The equation has infinitely many solutions before an initial condition is imposed.

Try it yourself

Does $\displaystyle y=t^2$ solve $\displaystyle y'=2y$ ?

Show solution and reasoning

No. Its derivative is $\displaystyle 2t$ , but $\displaystyle 2y=2t^2$ . Equality at isolated times does not establish a solution on an interval.