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- movement of an object
- the collision of two cars
- trajectories of planets

Imagine a particle that is projected horizontally (Gravity is neglected):
*x*
or
If we are interested in time then:

- Velocity is given by
- Acceleration is
- It is assumed that the drag is proportional to

- When

- When

Differential Equations which involve only one independent variable are called **Ordinary**.
In these equations *x* is the independent variable and *y* is the dependent variable.
For example :
Equations which involve two or more independent variables and partial differential coefficients with respect to them are called **Partial**. For example :

- When

- Equations that involve a second differential coefficient but none of higher orders is said to be
**Second**Order.

- When

- When

- When

For example :
**First order**
**Second order**
**Third order**

- The degree of an equation is the power of the highest differential coefficient once the equation has been made rational and integral as far as the differential coefficients are concerned. For example :

- When

- When

A **quadric function** is a function such as
where

If then
Now consider the equation
Therefore

- If A = 0 the graph is as above.
- If A = 1
- If A = -2

An **exponential function** is a function such as
where

Now consider the following equation:
This can be rearranged as:
The variables have now been separated and :
From which the explicit form is given by:

where *w* i aconstant
Integrating with respect to *x* gives
And so on until
Where *A*, *B*, *C* and *E* are all arbitrary constants

A Particular solution of
is given by
(Obtained by putting A,B,C,E = 0)
or

Example:

Problem

A cricket ball is thrown vertically upwards with a velocity of v ft/sec. The retardation is or . Find the maximum height reached (Y) and the time of flight to the vertex (T).
Prove that the Initial velocity u is given by:

Workings

The acceleration = -kv - g
To find the time of Flight T
thus
When t = 0 v = u
Thus
At the vertex t = T and v = 0 so
i.e.
For Height Y
But when y = 0, v = u so
At the Vertex v = 0

Solution

The flight time is:
Max Height is:

Example:

Problem

Basic trigonometrical examples

Workings

The reference page on Trigonometrical Formulae includes:-
Considering the first equation , this can be re-written as:-
Now if during the solution of a differential equation we arrive at :-
we can compare the right hand side with the right hand side of (5) and we can see that they are of
the same form but has been replaced by "3" and by "4". Clearly this
can not be correct as the Sine and Cosine can not have a value above unity but if we draw the
following right angled triangle.
Values of Sine and Cosine can be obtained which can be put into equation (5)
This can be re-arranged to satisfy the requirements of equation (6)

Example:

Problem

Solve

Workings

Using the D operator

Solution

Therefore the General Solution is given by:-

Computes the first and second derivatives of a function using the Taylor formula.

double | (double (*f)(double), double x, double h, double gamma = 1.0)taylor1[inline] |

double | (double (*f)(double), double x, double h, double gamma = 1.0)taylor2[inline] |

Computes the first and second derivatives of a function at multiple points.

std::vector<double> | (double (*f)(double), std::vector<double> &points, double h, double gamma = 1.0)taylor1_table |

std::vector<double> | (double (*f)(double), std::vector<double> &points, double h, double gamma = 1.0)taylor2_table |

First Order Differential Equations with worked examples

A guide to linear equations of second and higher degrees

This section contains worked examples of the type of differential equation which can be solved by integration

Solving Differential Equations using the D operator

The solution of homogeneous differential equations including the use of the D operator

Linear simultaneous differential equations

An Introduction to Partial Differential Equations