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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