y1 y2 y2 cos2 x 1 cos 2 x 2 sin2 x 1 cos 2 x 2 sin a b sin a cos b cos a sin b cos a b cos a cos b sin a sin b THEOREM Consider Initial Value Problem y p t y q t y g t y t0 y 0 y t 0 y 0 where p q and g are continuous on an open interval I that contains the point I 0 Then there is exactly one solution y t of this problem and the solution exists throughout the interval I WHAT IT MEANS EXACTLY one solution is defined where p q and g are continuous PROBLEM Find the longest interval in which the solution to initial value problem qwerty is certain to exist 1 Put qwerty in standard form 2 Find discontinuities in p q and g 3 Find the interval t 0 is in SUPERPOSITION THEOREM If y 1 and y 2 are solutions to y p t y q t y 0 Eq 2 then their linear combinations are solutions bigger matrix for higher order Can factor rows cols to Wronskian y1 simplify do one row col at a time Interchanging row cols negates the det Adding multiple of one row col to another doesn t change det Still ugly piece of s Use Abel s 3 2 3 y 1 and y2 are solutions to Eq 2 Assign initial value t 0 It is always possible to pick constants so that the linear combination satisfies the initial value problem if and only if W is not 0 att 0 3 2 4 If y1 and y2 are solutions to Eq 2 then the family of solutions created by their linear combinations include every solution of 2 if and only if there exists t0 where W is not 0 General Solution c1y1 t c2y2 t FUNDAMENTAL Solution a general solution where W is never 0 4 1 3 If solutions are lin Independent is a fundamental set of solutions LIN INDEPENDENCE Multiply your functions by k1 k2 set up a system Picking convenient values of t can sometimes do it Is there a nontrivial solution 3 2 6 If y u t iv t is a solution to 2 then u and v are also solutions Abel s Theorem If y1 and y2 are solutions of 2 then W c e p t dt Make sure there is no leading coefficient EULERS FORMULA eit cos t i sin t COMPLEX ROOTS 1 Find angle radius 2 Add 2 m to angle Plug into R ei 3 Multiply the exponent by the power you re trying to find 4 Euler s formula 5 Plug in m 0 1 2 3 according to how many roots there should be HOW TO SOLVE LINEAR HOMOGENOUS EQS 1 Solve for r1 and r2 2 Unequal y c1 er1 t c2 e r2 t 3 Equal y c1 er1 t c2 t er1 t 4 Complex y c1 e t cos t c2 e t sin t REDUCTION OF ORDER Used when p and q are nonconstant You know y1 Find y2 Set y as y v t y1 t Differentiate to find y and y Substitute into original equation and simplify The coefficient of v should be 0 Now let w v Solve the first order or separable for w w v so integrate to find v Substitute into y v t y1 t to find y t You forgot this last test monkey If y t contains a term that is a multiple of y1 drop it The other term is y2 The wronskian can be used to tell if y1 and y2 form a set of fundamental solutions FIRST ORDER Put it into the form y p t g t t e p t dt Multiply by t The left side is now a product rule that equals d Integrate and solve for y 3 5 2 Nonhomogeneous eq y p t y q t y g t Eq 1 Gen sol Can be written as y t c1 y1 t c 2 y2 t Y t y1 and y2 are solutions of the corresponding homogeneous equation g t 0 Y is a specific solution of nonhomogeneous 1 How to solve 1 Find general solution to homogeneous g t 0 2 Find Y t 3 Form the sm UNDETERMINED COEFFICIENTS 1 What form is Y t in For g t C ekt use Y t A ekt g t sin kt cos kt Use Y t A sin kt B cos kt g t ect cos kt ect sin kt Use Y t A ect cos kt B e ct sin kt g t polynomial Use polynomial of same degree Multiple terms Treat each term as its own problem Add the t y dt Y t s at the end Trick question Multiply the assumed form of Y t by t g is a complicated product Find an assumed Y by first finding the assumed Y s of the simple non exponential parts Multiply these together Multiply it by the exponential part at the end without the coefficient The assumed Y for g exp poly deg n trig would be exp poly deg n sin different poly deg n cos Match like terms to find A B C Use this method for exponentials sine cosine polynomials and sum products of the aforementioned If else use variation of parameters 4 3 UNDETERMINED COEFFICIENTS So one of your Y t is equal to one of the terms of yc Multiply the assumed form by a power to t high enough to prevent this from happening you can substitute into the original equation 2 Match coefficients to find your constants 3 Add to gen sol of homogeneous equation VARIATION OF PARAMETERS Nonhomogeneous eq not equal to 0 The general form is y c1y1 c2y2 Y t The general form of Y t is v1y1 v2y2 1 Solve corresponding homogeneous set to 0 obtain y1 and y2 2 Find Wronskian 3 Use following formulas v1 y2g W v2 y1g W 4 Integrate to find v1 and v2 5 Plug into general form of Y t then plug that into general form of solution VARIATION V4 4 1 Find solution of homo equation 2 Find Wronskian 3 Replace first col w 0 0 1 Find det This is W1 Repeat w rest of columns You got Y t now what Differentiate until 4 Each term of Yp is yn g t W n W Add this to Yc the sol of the homo eq Evil teacher You reduce the order and you get an EXACT equation These equations can be written in the form M x y N x y y 0 M and N satisfies M y x y N x x y There exists a function such that x x y M x y y x y N x y c is the solution To solve 1 Set up equations for x y 2 Integrate the first one x with respect to x to obtain The constant of integration is h y 3 Find …