Differential Equations

Calculus

First Order Differential Equations

Separating The Variables

Equations in the form can be solved with

Integrating Factor

An integrating factor is an that you multiply through by to solve differential equations in the form . The general integrating factor is but it can be other functions

Coupled First-Order Simultaneous Differential Equations

You can solve these equations by rearranging equation (1) to get then differentiate that and sub into equation 2 to get a 2nd order differential equation, then solve for , differentiate and plug into to get .
You could also solve for first

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Second Order Differential Equations

Homogeneous

A second order homogeneous differential equation is a linear differential equation with a 2nd derivative term that equals zero

It has the general solution where and are arbitrary constants, and and are constants to be determined. Plugging in into the differential equation, we get that and are solutions to the quadratic equation this is called the auxiliary equation.

This quadratic leads to 3 cases.

• The auxiliary equation has 2 real root.
• The auxiliary equation has 1 repeated root .
• The auxiliary equation has two complex conjugate roots . - This is equivalent to the first case with complex and

Non-Homogeneous

A second order non-homogeneous differential equation is a linear differential equation with a 2nd derivative term that equals some function of

It has the general solution . Where is the solution to the corresponding homogenous function and is a particular integral

Particular Integral

The particular integral is a specific function that satisfies the differential equation. To find the , assume that has the same form as , then plug it into the original equation to find the coefficients.

If the particular integral contains terms which form part of the complementary function you have to modify your so that no 2 terms in the general solution have the same form this is done by multiplying the by or

Form of	Form of

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Series Solutions of Differential Equations

You can use Taylor series to approximate solutions to Differential Equations that can’t be solved with other techniques. Suppose you have the equation and you have initial conditions Then you can calculate by substitution into . By successive differentiation of the original equation and substitution of previously found values, you can find values of Therefore the series expansion of is

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Modelling with Differential Equations

Simple Harmonic Motion

Simple harmonic motion (S.H.M) is motion in which acceleration is always towards a fixed point (the centre of oscillation), and proportional to the displacement (basically oscillating through a point).

S.H.M is modelled with , where is angular velocity

Damped and Forced Harmonic Motion

You can refine and make more accurate models for harmonic motion by adding an additional damping force.

The auxiliary equation leads to 3 cases:

• : Known as heavy damping - no oscillations as resistive force is large compared to restoring force
• : Known as critical damping - again no oscillations performed
• : Known as light damping - there are oscillations of which the amplitude decreases exponentially over time

For heavy and critical damping, the exact nature of the motion depends on initial conditions. For light damping, the period of observed oscillations can be calculated.