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Forming a Differential Equation

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Forming a Differential Equation - Lesson Summary

Building a differential equation whose general solution is given.

xy = ax2 + b/x

To find the differential equation, the solution is differentiated to the extent that the constants are eliminated.

By multiplying both the sides of the equation by x.

x(xy) = x(ax2 + b/x)   ⇒ x2y = ax3 + b

x2 dy/dx + 2xy = 3ax2 + 0

⇒ x dy/dx + 2y = 3ax ...........(I)

⇒ x d2y/dx2 + dy/dx + 2 dy/dx = 3a

⇒ x d2y/dx2 + 3 dy/dx = 3a

⇒ 3a = x d2y/dx2 + 3 dy/dx  .................. (II)

From (1) and (2)

⇒ x dy/dx + 2y = (x d2y/dx2 + 3 dy/dx)x

⇒ x dy/dx + 2y = x2 d2y/dx2 + 3x dy/dx

⇒ x2 d2y/dx2 + 3x dy/dx - x dy/dx - 2y = 0

⇒ x2 d2y/dx2 + 2x dy/dx - 2y = 0

Working rules to form a differential equation:

1. Observe the number of arbitrary constants present in the given general solution.

2. Find the derivative up to the order based on the number of constants.

Ex: x2 = 4ay (One constant: a)

Differential equation: x. dy/dx = 2y

x2/b2 + y2/a2 = 1 (Two constants: a and b)

Here the number of arbitrary constants is two, then differentiate the general solution twice.Then we get

Differential equation: xy d2y/dx2 + x(dy/dx)2 - y dy/dx = 0

3. Eliminate the arbitrary constants using the equations obtained.

xy = cex + be-x + x2 Constants: c,b

Differentiating both sides with respect to x:

x . dy/dx + y = cex - be-x + 2x……. (1)

Differentiating both sides with respect to x again:

x d2y/dx2 + dy/dx + dy/dx = cex + be-x + 2

⇒ x d2y/dx2 + 2. dy/dx = cex + be-x + 2 ….. (2)

⇒ x d2y/dx2 + 2. dy/dx = xy - x2 + 2 [Since, xy = cex + be-x + x2]

Required differential equation

x d2y/dx2 + 2. dy/dx - xy + x2 - 2 = 0.

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