Formation of Differential Equations – 01 (Single Arbitrary Constant)

In this article, we shall study to form a differential equation by eliminating a single arbitrary constant from the given relation.

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

∴  ye^x = c ………. (1)

Differentiating both sides w.r.t. x

y.e^x + e^x = 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(1) = 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(1) + 0 + 0 = 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(1) + 0 + 0 = 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 = x( – 2x) + 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

x + y= a

Substituting in equation (1)

∴  x² + y² = 2(x + y)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 = 2xy

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

x + y= a

Substituting in equation (1)

∴  x² + y² = 2(x + y)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

Substituting in equation (1)

∴ x²  = 2xy

∴ 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

= 0 + c(-1/x²) = -c/x²

c = – x²

Substituting in equation (1)

∴ y = x⁴()² – x.

∴  x⁴()² – x. – y = 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 = ax³ + 4

Solution:

y = ax³ + 4 …………….. (1)

Differentiating both sides w.r.t. x

= l

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:

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