Page 99 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Thus we see from above examples that if there is one constant, the resultant differential equation
is of first order. If there are two arbitrary constants the differential equation is of second order.
So from the above it is clear that nth order differential equation involves n arbitrary constant in
its solution.

Self Assessment

1. Find the differential equation for the Harmonic equation
y = A sin wt + B cos wt

Where A, B and w are constants.
2. Find the differential equation of the following curves

x 2 cy x e x
where c is an arbitrary constant.
3. Discuss the nature of the equation

2
d y dy 2 dy
a b c dy Q ( )
x
dx 2 dx dx
where a, b and c are constants.
4. Find the differential equation of the curve

y e x ( sin x c 2 cos )
x
c
1
5.3 Linear Ordinary Differential Equations of First Order

The most general first order differential equation can be put in the form
dy
y
f x , , 0 ...(1)
dx
dy
where f is any arbitrary function of x, y and . Various cases arise due to the nature of the
dx
dy
y
function f x , , 0 . In the following we consider a few of them with some examples.
dx
(A) The equation with separable variables. Here the equation can be put in the form
M(x) dx + N(y) dy = 0 ...(2)

where M(x) is a function of x and N(y) a function of y. So integrating (2) we have

y
M ( )dx N ( )dy a ...(3)
x
where a is an arbitrary constant

Example 1: Solve

2
y xy dx x dy 0

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