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

2. 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

3. Our ways to solve a diff eq A SINGLE DIFF EQ

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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 = 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)

10. The 3 possible solutions
Come from the Characteristic Equation

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

14. UNIT 4-SOLVING SYSTEMS OF EQUATIONS

15. 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

16. 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

17. 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