# Formation of Differential Equations – 01 (Single Arbitrary Constant)

### Formation of the differential equation by eliminating the arbitrary constant:

#### Example – 01:

• xy = c
• Solution:

Given xy = c ……….. (1)

Differentiating both sides w.r.t. x

x + y(1) = 0

∴   x + y = 0

This is the required differential Equation

#### Example – 02:

• xy2 = c2
• Solution:

xy2 = c2  ……….. (1)

Differentiating both sides w.r.t. x

x.2y + y2 (1) = 0

∴  2x + y = 0

This is the required differential equation

#### Example – 03:

• y = ce-x
• Solution:

y = ce-x

∴  yex = c ………. (1)

Differentiating both sides w.r.t. x

y.ex + ex = 0

∴  y + = 0

∴  + y = 0

This is the required differential Equation

#### Example – 04:

• x2 + y2 = a2
• Solution:

x2 + y2 = a2     ……………… (1)

Differentiating both sides w.r.t. x

2x + 2y = 0

∴  x + y = 0

This is the required differential Equation

#### Example – 05:

• y = ax + 2
• Solution:

y = ax + 2 ………… (1)

Differentiating both sides w.r.t. x

= a

Substituting in equation (1)

y =x + 2

∴  x – y + 2 = 0

This is the required differential equation

#### Example – 06:

• y = ax + a2 + 5
• Solution:

y = ax + a2 + 5 ……….. (1)

Differentiating both sides w.r.t. x

= a

Substituting in equation (1)

y = x + ()2 + 5

∴  ()2 + x – y + 5 = 0

This is the required differential equation

#### Example – 07:

• y = ax + 6a2 + a3
• Solution:

y = ax + 6a2 + a3……….. (1)

Differentiating both sides w.r.t. x

=a

Substituting in equation (1)

y = x + 6()2 + ()3

∴  ()3 +  6()2 + x – y = 0

This is the required differential equation

#### Example – 08:

• y = cx + x2
• Solution:

y = cx + x2 ……………… (1)

Differentiating both sides w.r.t. x

= c + 2x

∴  c = – 2x

Substituting in equation (1)

y = ( – 2x)x + x2

∴  y = x – 2x2 + x2

∴  x – x2 – y = 0

This is the required differential equation

#### Example – 09:

• (x – a) 2 + y2 = a2
• Solution:

(x – a) 2 + y2 = a2

∴  x2 – 2ax + a2 + y2 = a2

∴  x2 – 2ax + y2 = 0

∴  – 2ax + a2 + y2 = a2

∴  x2 + y2 = 2ax  ………… (1)

Differentiating both sides w.r.t. x

2x + 2y = 2a

Substituting in equation (1)

x2 + y2 = (2x + 2y)x

∴  x2 + y2 = 2x2 + 2xy

∴  2xy + x2 – y2 = 0

This is the required differential equation

#### Example – 10:

• y2 = 4ax
• Solution:

y2 = 4ax ……….. (1)

Differentiating both sides w.r.t. x

2y = 4a

Substituting in equation (1)

y2 = 2yx

∴  y = 2x

∴  2x – y = 0

This is the required differential equation

#### Example – 11:

• x2 + y2 = 2ax
• Solution:

x2 + y2 = 2ax    ……………. (1)

Differentiating both sides w.r.t. x

2x + 2y= 2a

Substituting in equation (1)

x2 + y2 = (2x + 2y)x

∴  x2 + y2 = 2x2 + 2xy

∴  2xy + x2 – y2 = 0

This is the required differential equation

#### Example – 12:

• x2  = 4ay
• Solution:

x2  = 4ay  …………. (1)

Differentiating both sides w.r.t. x

2x = 4a

∴  4a = (2x)/()

Substituting in equation (1)

∴  x = 2y

∴  x – 2y = 0

This is the required differential equation

#### Example – 13:

• (y – b)2 + x2 = b2
• Solution:

(y – b) 2 + x2 = b2

∴  y2 – 2by + b2 + x2 = b2

∴  y2 – 2by + x2 = 0

∴  x2 + y2 = 2by  ………… (1)

Differentiating both sides w.r.t. x

2x + 2y = 2b

Substituting in equation (1)

∴  x2 + y2 = 2xy + 2y2

∴  x2 – y2 – 2xy = 0

∴  (x2 – y2) – 2xy = 0

This is the required differential equation

#### Example – 14:

• y = c2 + c/x
• Solution:

y = c2 + c/x  ………… (1)

Differentiating both sides w.r.t. x

= c(-1/x2)

∴  c = – x2

Substituting in equation (1)

∴  x = x4()2 – x

∴  x4()2 – x – x = 0

This is the required differential equation

#### Example – 15:

• ex + c ey = 1
• Solution:

ex + c ey = 1 …… (1)

Differentiating both sides w.r.t. x

ex + c ey = 0

∴  c ey = – ex

Substituting in equation (1)

This is the required differential equation

#### Example – 16:

• y = ax3 + 4
• Solution:

y = ax3 + 4 …………….. (1)

Differentiating both sides w.r.t. x

= a.3x2

∴  a = ()/(3x2)

Substituting in equation (1)

∴  3y = x + 12

∴  x – 3y + 12 = 0

This is the required differential equation

#### Example – 17:

• ex + ey = k ex + y
• Solution:

ex + ey = k ex + y

Differentiating both sides w.r.t. x

This is the required differential equation

#### Example – 18:

• ex + key = 1
• Solution:

ex + key = 1 …….. (1)

Differentiating both sides w.r.t. x

ex + key = 0

∴  key = – ex

Substituting in equation (1)

This is the required differential equation

#### Example – 19:

• y  = ecx
• Solution:

y  = ecx

∴  log y  = log ecx

∴  log y  = cx log e = cx (1)

∴  log y = cx ……….. (1)

Differentiating both sides w.r.t. x

(1/y) = c

Substituting in equation (1)

This is the required differential equation