# Applications of Differential equations

by Squires
Tags: applications, differential, equations
 P: 743 This is a differential equation, and so we want to apply differential techniques to it. In particular, might I suggest separation of variables? Namely, if you have an equation $$\frac{dM}{dt} = aM$$ try "multiplying" everything by $$\frac{dt}{M}$$. You'll then be in a position to integrate to find a solution. Don't forget your constant (hint for $M_0$.)
 P: 743 Applications of Differential equations No problem. While mathematically it pains me to say this, you should treat $\frac{dM}{dt}$ as a fraction so that you can separate the dM and the dt components. Now what do these terms look like? They look like integration terms! $$\int f(x) \underbrace{dx}_{\uparrow}$$ Here the dx terms tell us that we're integrating with respect to x. Hence our goal with $$\frac{dM}{dt} = a M$$ will be to move all the "M" terms to one-side of the equal sign, and all the "t" terms to the other. Multiply both sides by dt and divide both sides by M, as if you were cancelling denominators in fractions \begin{align*}\left( \frac{\cancel{dt}}{M} \right) \frac{dM}{\cancel{dt}} &= \left( \frac{dt}{\cancel M} \right)(a \cancel M) \\ \frac{dM}M &= a dt \end{align*} Now these look like integrations right? So trying throwing an integral sign out front $$\int \frac1M dM = a \int dt$$ Try integrating these equations (not forgetting your constant of integration!) and solve for M.