### 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:

- xy
^{2}= c^{2} **Solution:**

xy^{2} = c^{2} ……….. (1)

Differentiating both sides w.r.t. x

x.2y + y^{2} (1) = 0

∴ 2x + y = 0

This is the required differential equation

**Example – 03:**

- y = ce
^{-x} **Solution:**

y = ce^{-x}

∴ ye^{x} = c ………. (1)

Differentiating both sides w.r.t. x

y.e^{x} + e^{x} = 0

∴ y + = 0

∴ + y = 0

This is the required differential Equation

#### Example – 04:

- x
^{2}+ y^{2}= a^{2} **Solution:**

x^{2} + y^{2} = a^{2 } ……………… (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 + a
^{2}+ 5 **Solution:**

y = ax + a^{2} + 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 + 6a
^{2}+ a^{3} **Solution:**

y = ax + 6a^{2} + a^{3}……….. (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 + x
^{2} **Solution:**

y = cx + x^{2} ……………… (1)

Differentiating both sides w.r.t. x

= c + 2x

∴ c = – 2x

Substituting in equation (1)

y = ( – 2x)x + x^{2}

∴ y = x – 2x^{2} + x^{2}

∴ x – x^{2} – y = 0

This is the required differential equation

#### Example – 09:

- (x – a)
^{ 2}+ y^{2}= a^{2} **Solution:**

(x – a)^{ 2} + y^{2} = a^{2}

∴ x^{2} – 2ax + a^{2} + y^{2} = a^{2}

∴ x^{2} – 2ax + y^{2} = 0

∴ – 2ax + a^{2} + y^{2} = a^{2}

∴ x^{2} + y^{2} = 2ax ………… (1)

Differentiating both sides w.r.t. x

2x + 2y = 2a

Substituting in equation (1)

x^{2} + y^{2} = (2x + 2y)x

∴ x^{2} + y^{2} = 2x^{2} + 2xy

∴ 2xy + x^{2} – y^{2} = 0

This is the required differential equation

#### Example – 10:

- y
^{2}= 4ax **Solution:**

y^{2} = 4ax ……….. (1)

Differentiating both sides w.r.t. x

2y = 4a

Substituting in equation (1)

y^{2} = 2yx

∴ y = 2x

∴ 2x – y = 0

This is the required differential equation

#### Example – 11:

- x
^{2}+ y^{2}= 2ax - Solution:

x^{2} + y^{2} = 2ax ……………. (1)

Differentiating both sides w.r.t. x

2x + 2y= 2a

Substituting in equation (1)

x^{2} + y^{2} = (2x + 2y)x

∴ x^{2} + y^{2} = 2x^{2} + 2xy

∴ 2xy + x^{2} – y^{2 }= 0

This is the required differential equation

#### Example – 12:

- x
^{2}= 4ay - Solution:

x^{2} = 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}+ x^{2}= b^{2} **Solution:**

(y – b)^{ 2} + x^{2} = b^{2}

∴ y^{2} – 2by + b^{2} + x^{2} = b^{2}

∴ y^{2} – 2by + x^{2} = 0

∴ x^{2} + y^{2} = 2by ………… (1)

Differentiating both sides w.r.t. x

2x + 2y = 2b

Substituting in equation (1)

∴ x^{2} + y^{2} = 2xy + 2y^{2}

∴ x^{2} – y^{2} – 2xy = 0

∴ (x^{2} – y^{2})^{} – 2xy = 0

This is the required differential equation

**Example – 14:**

- y = c
^{2}+ c/x **Solution:**

y = c^{2} + c/x ………… (1)

Differentiating both sides w.r.t. x

= c(-1/x^{2})

∴ c = – x^{2}

Substituting in equation (1)

∴ x = x^{4}()^{2} – x

∴ x^{4}()^{2} – x – x = 0

This is the required differential equation

#### Example – 15:

- e
^{x}+ c e^{y}= 1 **Solution:**

e^{x} + c e^{y} = 1 …… (1)

Differentiating both sides w.r.t. x

e^{x} + c e^{y} = 0

∴ c e^{y} = – e^{x}

Substituting in equation (1)

This is the required differential equation

#### Example – 16:

- y = ax
^{3}+ 4 - Solution:

y = ax^{3} + 4 …………….. (1)

Differentiating both sides w.r.t. x

= a.3x^{2}

∴ a = ()/(3x^{2})

Substituting in equation (1)

∴ 3y = x + 12

∴ x – 3y + 12 = 0

This is the required differential equation

#### Example – 17:

- e
^{x}+ e^{y}= k e^{x + y} **Solution:**

e^{x} + e^{y} = k e^{x + y}

Differentiating both sides w.r.t. x

This is the required differential equation

#### Example – 18:

- e
^{x}+ ke^{y}= 1 - Solution:

e^{x} + ke^{y} = 1 …….. (1)

Differentiating both sides w.r.t. x

e^{x} + ke^{y} = 0

∴ ke^{y} = – e^{x}

Substituting in equation (1)

This is the required differential equation

#### Example – 19:

- y = e
^{cx} **Solution:**

y = e^{cx}

∴ log y = log e^{cx}

∴ 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