All the way from Unit 1 to Unit 4 Diff Eq Final Review All the way from Unit 1 to Unit 4

Easiest Point-Verifying a diff eq This simply plug and chug, making sure both sides are equal Take the 1st and 2nd derivative and plug all into the diff eq Ex: Verify 𝑦 𝑡 = 6 5 − 6 5 𝑒 −20𝑡 𝑖𝑠 𝑎 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 𝑑𝑦 𝑑𝑡 +20𝑦=24 Take the derivatives, 1st : y’(t)=24 𝑒 −20𝑡 , no need for the 2nd Plug into the original equation 24 𝑒 −20𝑡 +20( 6 5 − 6 5 𝑒 −20𝑡 )=24 Simplify: 24 𝑒 −20𝑡 +24−24 𝑒 −20𝑡 )=24 So 24=24

Our ways to solve a diff eq A SINGLE DIFF EQ

These are The main Things to know Solving a diff eq Unit 1 Homogenous Phase Portraits Non-Homogenous Linear and Separation of Variables Exact Equations Unit 2 No RHS (It is = to 0) 3 possible solutions RHS (There is a driving force) Method of Undetermined Coefficents Variation of Parameters Unit 3 Solving through laplace transforms Unit Step Functions Dirac Delta Functions These are The main Things to know

Unit 1-The Phase Portrait If the equation is y(assuming the dependent variable) (squared, cubed, multiplied, plus) =0 Factor-Find zeros Graph the zeros on the number line Figure out the signs in between, find attractors and repellers Now graph the solution, attractors and repellers Repeller Attractor Attractor Repeller

Unit 1- 3 forms of solving the equation Separation of variables When u can move everything with x to one side and everything with y to the other side, then take the integral of both sides After taking the integral Explicit (solve for y) Implicit (Don’t worry Linear When the equation is in a very distinct form or can be manipulated into that form Exact See if u can change into that formula Then check to see if it is exact Than take 2 integrals

Separation of variables Know how to do IBP, u-sub and basic trig integrals Process Move everything with x and everything with y to each side Leave 𝑑𝑦 𝑑𝑥 on the side with the y’s Multiply by dx 𝑑𝑦 𝑑𝑥 𝑦 𝑠𝑡𝑢𝑓𝑓 = 𝑥 𝑠𝑡𝑢𝑓𝑓 = 𝑑𝑦 𝑦 𝑠𝑡𝑢𝑓𝑓 =(𝑑𝑥)(𝑥 𝑠𝑡𝑢𝑓𝑓) Take the integral of both sides 𝑦 𝑠𝑡𝑢𝑓𝑓 𝑑𝑦 = 𝑥 𝑠𝑡𝑢𝑓𝑓 𝑑𝑥 If it asks for the implicit leave it as is, if it asks for the explicit solution you have to solve for y in terms of x

The linear Case Where p(x) and q(x) are y as a function of x Meaning you will only see x’s Get into the following Form in you can 2. Find the integrating factor 3. Multiply by the integrating factor 𝑑 𝑑𝑥 𝑦 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟 =RHS(Integrating factor) 4. Integrate both sides 𝑑 𝑑𝑥 𝑦 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟 𝑑𝑥= 𝑅𝐻𝑆 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟 𝑑𝑥 5. The LHS simply becomes (y)(Integrating factor)= 𝑹𝑯𝑺 𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒕𝒊𝒏𝒈 𝒇𝒂𝒄𝒕𝒐𝒓 𝒅𝒙 6. Now take the integral and solve for y if it asks for the explicit solution

The exact case Where M and N can be functions of x and y Remember when taking partials, treat the Other variable as a constant Ex: 𝑑 𝑑𝑥 2𝑥 =𝑥 so 𝑑 𝑑𝑥 𝑥𝑦 =𝑥 𝑑 𝑑𝑥 2 =0 so 𝑑 𝑑𝑥 𝑦 =0 Notice very distinct difference Exact equations look like Make sure the equation is exact by checking Take 2 integrals (Don’t forget the plus C) In this case 𝑀𝑑𝑥 =𝑆𝑡𝑢𝑓𝑓+𝒇(𝒚) And 𝑁𝑑𝑦 =𝑆𝑡𝑢𝑓𝑓+𝒈(𝒙) 4. Now go on hunt and combine the 2 integrals a. Pick one of the integrals and grab f(x) or g(y) from the other Same story on partials here 3𝑥𝑑𝑥 =3( 𝑥 2 2 ) 𝑥𝑦𝑑𝑥 =𝑦( 𝑥 2 2 ) Need 𝑀𝑑𝑥 and 𝑁𝑑𝑦 Ex: That’s say you get the following out of the integrals 𝑓 𝑥,𝑦 = 𝑥 2 𝑦+ 𝑦 2 +𝒇 𝒙 𝑓 𝑥,𝑦 = 𝑥 2 𝑦+𝐥𝐧 𝐱 +g(y) Ans: 𝑓 𝑥,𝑦 = 𝑥 2 𝑦+ 𝑦 2 +ln⁡(𝑥)

The 3 possible solutions Come from the Characteristic Equation

Variation of parameters For Martin-Formulas is given Kumudu-The Determinants

UNIT 4-SOLVING SYSTEMS OF EQUATIONS

Write in general matrix form Apply the identity Or use… Write in the general solution form (This assumes 2 real roots) Use the following idea to solve for the coefficients Apply for both the eigenvalues to solve for the 2 eigen vectors When you solve this, you will get 2 equations, you can pick either one, and This will tell you a relationship between k1 and k2, set k1=0 and solve for k2 Put into the vector 𝐾 1 𝐾 2 , this is the eigenvector to correspond with the 𝜆

𝑑𝑒𝑡𝐴>0, 𝑡𝑟𝐴<0 (𝑡𝑟𝐴) 2 <4𝑑𝑒𝑡𝐴 𝑑𝑒𝑡𝐴>0, 𝑡𝑟𝐴>0 (𝑡𝑟𝐴) 2 <4𝑑𝑒𝑡𝐴 𝑑𝑒𝑡𝐴>0, 𝑡𝑟𝐴>0 (𝑡𝑟𝐴) 2 >4𝑑𝑒𝑡𝐴 𝑑𝑒𝑡𝐴>0, 𝑡𝑟𝐴<0 (𝑡𝑟𝐴) 2 >4𝑑𝑒𝑡𝐴 𝑑𝑒𝑡𝐴<0

Unit 1 You can only use the method for linear if the equation can be put into that form In solving linear, remember to multiply the and RHS and LHS by the integrating factor You can factor out, divide and multiply to separate variables, but be careful! 𝑦 2 𝑥−𝑦, 𝑦𝑜𝑢 𝒄𝒂𝒏𝒏𝒐𝒕 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑢𝑡 𝑡ℎ𝑒 𝑦 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑦 2 𝑥−𝑦=𝑦 𝑥𝑦−1 , 𝑥,𝑦 still trapped together Unit 2 If the Guess matches the general solution, modify by multiplying by x Remember to keep the nature (exponent in a e, argument in a sin or cos), when making a guess on the RHS For variation of parameters, it should be a simple integral you are left with Unit 3 Remember the tricks for unit step functions and dirac delta functions Both unit step functions and dirac delta functions in the original should lead to an answer with a unit step Remember laplace in the beginning, algebraically rearrange, then take the inverse laplace Unit 4 Really study the chart corresponding to the different possibilities Remember the difference between a stable and unstable node and what makes each Know the process of how to get the eigenvectors and don’t forget they are vectors, so u need both components, set 1 comp. = to 1, use relation to find the other