Differential equations

Question 1

First order differentiatial equations process

Answer 1

Ensure that there is nothing multiplying dy/dx if there is divide it out.
Find what is multiplying the y.(p(x))
Find integrating factor (u(x) by putting integral of what is multiply y to be e to the power if it. u(x)= e^§p(x)dx
Solve.
Put into equation u(x)y= §u(x)*Q(x)dx where Q(x) is our original answer.
Solve

Question 2

e^lnx =

Answer 2

x

Question 3

Homogeneous 2nd order differentiation process for general solution

Answer 3

Write out equation
Find auxiliary equation by replacing dy/dx examples by D.
Solve now quadratic equation to put into complementary function.

Question 4

Complementary function for two real different roots

Answer 4

y=Ae^root 1x+ Be^root2x

Question 5

Complementary function for one real root

Answer 5

y=(A+Bx)e^rootx

Question 6

Complementary function for no real roots

Answer 6

y = e^ax(Acosbx+Bsinbx)

a is number alone
b is number that * i

Question 7

Solving auxiliary equations for no real roots

Answer 7

Use quadratic formula, solve but when it gives a negative power solve it by using i = square root of -1 to get answer that will look like
4+-3i for example

Question 8

How to solve non homogeneous second order differentiatial equations

Answer 8

Do what is required for homogeneous assuming LHS =0 Find particular integral (pi)
First write out y = basic formula for RHS
Then solve first and second order differentiations for them.
Substitute these in when required.
Simplify equation.
Use simultaneous equations or equivalence to determine what a and b are equal to.
Rewrite PI with a and b subbed in.
Add P.I. to LHS in general solution.

Question 9

Equivalence

Answer 9

When there is only one term that can equal a thing on the other side so it must be equal
Ie.
3a-4b-4ax = 4x + 5
-4ax = 4x
3a- 4b = 5

Question 10

PI for 4x^2 + 3x +7

Answer 10

ax^2 + bx + c

Question 11

PI for 4x^2 + 7

Answer 11

ax^2 + bx + c

Question 12

PI for e^3x

Answer 12

ae^3x

Question 13

PI for 3sin2x

Answer 13

asin2x + bcos2x

Question 14

PI for e^x

Answer 14

ae^x (so long as complementary function doesn’t already contain e^x if so multiply by x)

Question 15

General solution

Answer 15

Involves constants eg. + c

Question 16

Particular solution

Answer 16

We work out the constants at put those into the final answer

Question 17

How can we tell if a differential equation is separable

Answer 17

If we can move all the y terms to one side of the equation and all the x terms to the other.

Question 18

How to solve separable differential equations

Answer 18

Integrate both sides.
Try to get y on its own.
If we’re multiplying or dividing both sides this will change +c so change to +C

Question 19

How to get y on its own when it’s lny

Answer 19

put it as the power of e as e^lny = y, must do to both sides.
When we do it to other sides +c—> A which multiplies e