MATH 220 Lecture 3: Seperable Equations

• School: University of Maryland
• Department: Mathematics
• Course: MATH 220
• Professor: Heyrim Cho

Document Summary

This is our first look at non-linear first order differential equations. We will take a 3 step: rewrite into the form (cid:4666)(cid:1877)(cid:4667)(cid:1856)(cid:1877)=(cid:4666)(cid:1876)(cid:4667)(cid:1856)(cid:1876, integrate both sides: (cid:4666)(cid:1877)(cid:4667)(cid:1856)(cid:1877)= (cid:4666)(cid:1876)(cid:4667)(cid:1856)(cid:1876: solve for the general or explicit solution. There"s not much left to say except let"s work through a few examples, so let"s work through a few examples! The integral of 1/y is the natural log function and the integral of (cid:889)(cid:1876)(cid:2871) is (cid:2875)(cid:2872)(cid:1876)(cid:2872). And now we solve for (cid:1877) by taking the exponential of both sides. Let"s work through one with a condition: (cid:1858) (cid:4666)(cid:1876)(cid:4667)=(cid:2870)(cid:3051)(cid:3120) (cid:3051) (cid:2869) (cid:2870)(cid:3052)(cid:3119) (cid:2874) , (cid:1877)(cid:4666)(cid:883)(cid:4667)=(cid:883)(cid:887) Ok, first we rewrite it into a proper form. And we solve for (cid:1855) (cid:4666)(cid:884)(cid:1877)(cid:2871) (cid:888)(cid:4667) (cid:1856)(cid:1877)=(cid:4666)(cid:884)(cid:1876)(cid:2872) (cid:1876) (cid:883)(cid:4667) (cid:1856)(cid:1876) And now that we have the constant, we can plug back into the equation. (cid:883)(cid:884)(cid:1877)(cid:2872) (cid:888)(cid:1877)=(cid:884)(cid:887)(cid:1876)(cid:2873) (cid:883)(cid:884)(cid:1876)(cid:2870) (cid:1876)+(cid:884)(cid:887)(cid:884)(cid:884)(cid:885). (cid:888) (cid:883)(cid:884)(cid:1877)(cid:2872) (cid:888)(cid:1877) (cid:884)(cid:887)(cid:1876)(cid:2873)+(cid:883)(cid:884)(cid:1876)(cid:2870)+(cid:1876) (cid:884)(cid:887)(cid:884)(cid:884)(cid:885). (cid:888)=(cid:882) Unfortunately, we can"t find a solution to this one.