Differential Equations Flashcards
What is the order in a differential equation
- The highest derivative which occours in the equation.
- E.g d^2y / dt^2 would be a second order differential equation
What is the degree of a differential equation
- The power which the highest derivative is raised
- Once fractional powers have been removed
What does it mean for a differential equation to be linear
- An equation is linear if it is linear in the dependant variable (y) and all of its derivatives (dy/dx) etc
- In all other cases the differential equation is said to be non-linear
What is a general solution of a DE
- The general solution contains the same number of arbitary constants as the DE’s order
What is a Particular solution of a DE
- A solution with no arbitrary constants
What is an ordinary differential equation
- Is any relation between a function of a single variable and its derivatives
How can you solve First Order DE’s
- Can be solved by seperation of variables
- Rearrange equation dependant on one side and independant on the other
What is the form of a type 1, First order DE
What is the form of a 1st order type 2 DE
- f(x,y) and g(x,y) MUST be the same degree
How to solve a 1O Type 2 DE
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What is the form of a first order type 3 DE
What is the critacal feature for a Type 3 1st order DE
ie δP / δΥ = δQ / δx
How to solve a first order type 3
What is a useful expression when dealing with first order type 3/4
What is the general form of a type 4 differential equation
How to solve a type 4 first order DE
- Multiply by an intergrating factor
- Solve as a normal type 3
What is a second order DE
- A differential equation with d^2/dx^2, dy/dx, y, x
What is the general form of a second order type 1 DE
How to solve a second order type 1 DE
- Form and auxillary quadratic
- Solve this quadratic
What is the key identity for 2nd order type 1
- y = e^mx
- Therefore this turns the equations into the form am^2 + bm + c = 0
- This is the auxiallary quadratic equation
What are the three cases for a solution to a type 1 second order DE
- If the answers to the auxillary quadratic are
- Case 1: Distint (i.e m1 = 2, m2 = 3)
- Case 2: Equal (i.e m1 = 2, m2 = 2)
- Case 3: Compelx ( a+- bj)
Whats the solution for a 2O T1 C1 DE
- Distinct roots
- y = Ae^m1x + Be^m2x
Whats the solution for a 2O T1 C2 DE
- y = Ae^m1x + xBe^m2x
- Equal roots
Whats the soluton for a 2O T1 C3 DE
- Complex roots (a +- bj)
- y = e^ax [Acos(bx) + Bsin(bx)]
How to find the exact solution to a 2O T1 DE
- First sub the x and y conditions into the general solution to get an eqaution in the form of A and B
- Differentaite the general solution, then subsitute the given y and dy/dx conditions
- Solve these two equations simultainiously to yield A&B
What is the form of a 2O T2 DE
