Solution Of First Order Equations Worksheets With Answers

Solution Of First Order Equations Worksheets With Answers - S. Ghorai

S. Ghorai

Lecture III

Solution of ﬁrst order equations

Separable equations

These are equations of the form

y = f (x)g(y)

Assuing g is nonzero, we divide by g and integrate to ﬁnd

f (x)dx + C

g(y)

What happens if g(y) becomes zero at a point y = y

Example 1. xy = y + y

Solution: We write this as

+ C

= ln x + C

ln y

ln(1 + y) = ln x + C

y + y

1 + y

Note: Strictly speaking, we should write the above solution as

ln y

ln 1 + y = ln x + C

When we wrote the solution without the modulas sign, it was (implicitly) assumed

that x > 0, y > 0. This is acceptable for problems in which the solution domain is not

given explicitly. But for some problems, the modulas sign is necessary. For example,

consider the following IVP:

xy = y + y

y( 1) =

Try to solve this.

Reduction to separable form

2.1

Substitution method

Let the ODE be

y = F (ax + by + c)

Suppose b = 0. Substituting ax + by + c = v reduces the equation to a separable form.

If b = 0, then it is already in separable form.

Example 2. y = (x + y)

Solution: Let v = x + y. Then we ﬁnd

v = v

+ 1

tan

v = x + C

x + y = tan(x + C)

Solution Of First Order Equations Worksheets With Answers - S. Ghorai