Introduction to Differential Equations

Definition:

A differential equation is an equation containing an unknown function and its derivatives.

32 xdx

dy

032

2

aydx

dy

dx

yd

364

3

3

ydx

dy

dx

yd

Examples:.

y is dependent variable and x is independent variable, and these are ordinary differential equations

1.

2.

3.

Ordinary Differential Equations

Order of Differential Equation

The order of the differential equation is order of the highest derivative in the differential equation.

Differential Equation ORDER

32 x

dx

dy

0932

2

ydx

dy

dx

yd

364

3

3

ydx

dy

dx

yd

1

2

3

Degree of Differential Equation

Differential Equation Degree

032

2

aydx

dy

dx

yd

364

3

3

ydx

dy

dx

yd

0353

2

2

dx

dy

dx

yd

1

1

3

The degree of a differential equation is power of the highest order derivative term in the differential equation.

Solution of Differential EquationSolution of Differential Equation

• Solving a differential equation means finding an equation with no derivatives that satisfies the given differential equation.

• Solving a differential equation always involves one or more integration steps.

• It is important to be able to identify the type of differential equation we are dealing with before we attempt to solve it.

Direct Integration

Direct Integration

Separation of Variables

Separation of Variables•Some differential equations can be solved by the method of separation of variables (or "variables separable") . This method is only possible if we can write the differential equation in the form

A(x) dx + B(y) dy = 0,where A(x) is a function of x only and B(y) is a function of y only.

•Once we can write it in the above form, all we do is integrate throughout, to obtain our general solution.

Differential Equation Chapter 1

16

Differential Equation Chapter 1 18

y = A. ln (x)

Particular SolutionsOur examples so far in this section have involved some constant of integration, K.We now move on to see particular solutions, where we know some boundary conditions and we substitute those into our general solution to give a particular solution.

24

Homogeneous Equations

Homogeneous Equations• Sometimes, the best way of solving a DE is a to use a change of variables that will put the DE into

a form whose solution method we know.

• We now consider a class of DEs that are not directly solvable by separation of variables, but,

through a change of variables, can be solved by that method.

If a function M(x, y) has the property that :M(tx, ty) = tnM(x, y), then we say the function M(x, y) is a ho-mogeneous function of degree n

A differential equation

M(x, y)dx + N(x, y)dy = 0is said to be a homogeneous differential equation if both functions M(x, y) and N(x, y) arehomogeneous functions of (the same) degree n. In this case, we can represent the functions as

M(x, y) = xnM(1, y/x)N(x, y) = xnN(1, y/x)

Answers Page 9 -10

Linear Equations – Use of Integrating Factors