Midterm 1 REVIEW 2 1 Linear Equations get y g d dx and integrate a first order DE is linear if it fits the for 1 a 1 x y a 0 x y bx 2 Standard form y p x y g x e p x dx multiply everything by y p x y g x 2 2 First order separable differential equations Homogeneous DE f x y g y x 2 3 Modeling w first order equations 1 Define IV and DV any parameters and identify their units 2 List any assumptions or physical laws used to relate the quantities 3 use physical laws to formulate the DE 4 verify that the units within the DE are consistent 5 solve and analyze 2 4 Differences between Linear and Nonlinear Theorem If the functions p x and g x are continuous on an open interval I containing the point xo then there exists a unique function y x that satisfies y p x y g x y xo yo on the interval I for any choice of yo Bernoulli Equations Bernoulli DE are DE of the form y x y g x y n where n is any real number Steps make sure in Bernoulli equation form Divide by yn Place v y1 n Transformed DE is v x y1 n g x 1 1 n 2 6 Exact Differential Equations The Differential equation M x y N x y 0 is called exact if there exists a function F x y dy dx such that dF dx M dF dy N can be rewritten as F x y 0 integrate with respect to x d dx m F Mdx dF N F N dx dx It is sometimes possible to make a DE that is not exact exact by multiplying by x y dF dy 2 for all x y for all real numbers F x y c Test for Exactness dM dy dN dx 1 F M dx 2 solve for dF dy There exists a function F such that 1 3 4 5 If DE is exact then m N d dy d dx d dy M dM dy d dx N dN dx 6 Choose which equation eliminates a variable dM dy dN dx a x d dx dM dy dN dx N N b y d dx 7 Pick c 1 8 Multiply by 9 Then find M and N the first order DE f x y is autonomous if the independent variable does not 2 5 Autonomous Equations dy dx explicitly f y dy dx