MATH 246 Lecture Notes - Lecture 7: Differential Form, Integrating Factor

• School: University of Maryland
• Department: Mathematics
• Course: MATH 246
• Professor: antoinemellet

Document Summary

Math246 lecture 7 differential form & integrating factor. Differential form is exact if there exists h(x,y) such that then h(x,y) = c is the implicit g. s of the d. e. Theorem : the differential form is exact if and only if (cid:1877)(cid:1839) (cid:4666)(cid:1876),(cid:1877)(cid:4667)= (cid:1876)(cid:1840)(cid:4666)(cid:1876),(cid:1877)(cid:4667) Solve the initial value problem and find the general solution. Differentiate m(x,y) with respect to only y and also differentiate n(x,y) with respect to only: you will then get : They are equal, so the form is exact. However, we can also write the differential equation as. M(x,y)dx + n(x,y)dy = 0 (cid:1856)(cid:1877)(cid:1856)(cid:1876)= (cid:1839)(cid:4666)(cid:1876),(cid:1877)(cid:4667) (cid:1840)(cid:4666)(cid:1876),(cid:1877)(cid:4667) (cid:1856)(cid:3051)=(cid:1839) (cid:1856)(cid:3052)=(cid:1840) (cid:1856)(cid:1877)(cid:1856)(cid:1876) + (cid:1857)(cid:3051)(cid:1877)+(cid:884)(cid:1876) (cid:884)(cid:1877)+ (cid:1857)(cid:3051)=(cid:882) (cid:1877)(cid:4666)(cid:882)(cid:4667)=(cid:882) (cid:4666)(cid:1857)(cid:3051)(cid:1877)+(cid:884)(cid:1876)(cid:4667)(cid:1856)(cid:1876)+ (cid:4666)(cid:884)(cid:1877)+ (cid:1857)(cid:3051)(cid:4667)(cid:1856)(cid:1877)=(cid:882) (cid:1877)(cid:1839) (cid:4666)(cid:1876),(cid:1877)(cid:4667)= (cid:1876)(cid:1840)(cid:4666)(cid:1876),(cid:1877)(cid:4667)= (cid:1857)(cid:3051) (cid:1857)(cid:3051)(cid:1877)+(cid:884)(cid:1876) (cid:884)(cid:1877)+ (cid:1857)(cid:3051)(cid:1856)(cid:1876)+(cid:1856)(cid:1877)=(cid:882) (cid:1876)(cid:1840)=(cid:882) (cid:1877)(cid:1839) (cid:882) They are not equal, so the form is not exact. However, since it can be arranged in such a way that the equation can be equal. But if it is not equal, we will have to figure out the integrating factor.