Summary

Videos

References

Building a differential equation whose general solution is given.

xy = ax^{2} + 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(ax^{2} + b/x) â‡’ x^{2}y = ax^{3} + b

x^{2} dy/dx + 2xy = 3ax^{2} + 0

â‡’ x dy/dx + 2y = 3ax ...........(I)

â‡’ x d^{2}y/dx^{2} + dy/dx + 2 dy/dx = 3a

â‡’ x d^{2}y/dx^{2} + 3 dy/dx = 3a

â‡’ 3a = x d^{2}y/dx^{2} + 3 dy/dx .................. (II)

From (1) and (2)

â‡’ x dy/dx + 2y = (x d^{2}y/dx^{2} + 3 dy/dx)x

â‡’ x dy/dx + 2y = x^{2} d^{2}y/dx^{2} + 3x dy/dx

â‡’ x^{2} d^{2}y/dx^{2} + 3x dy/dx - x dy/dx - 2y = 0

â‡’ x^{2} d^{2}y/dx^{2} + 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: x^{2} = 4ay (One constant: a)

Differential equation: x. dy/dx = 2y

x^{2}/b^{2} + y^{2}/a^{2} = 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 d^{2}y/dx^{2} + x(dy/dx)^{2} - y dy/dx = 0

3. Eliminate the arbitrary constants using the equations obtained.

xy = ce^{x} + be^{-x} + x^{2} Constants: c,b

Differentiating both sides with respect to x:

x . dy/dx + y = ce^{x} - be^{-x} + 2xâ€¦â€¦. (1)

Differentiating both sides with respect to x again:

x d^{2}y/dx^{2} + dy/dx + dy/dx = ce^{x} + be^{-x} + 2

â‡’ x d^{2}y/dx^{2} + 2. dy/dx = ce^{x} + be^{-x} + 2 â€¦.. (2)

â‡’ x d^{2}y/dx^{2} + 2. dy/dx = xy - x^{2} + 2 [Since, xy = ce^{x} + be^{-x} + x^{2}]

Required differential equation

x d^{2}y/dx^{2} + 2. dy/dx - xy + x^{2} - 2 = 0.

Building a differential equation whose general solution is given.

xy = ax^{2} + 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(ax^{2} + b/x) â‡’ x^{2}y = ax^{3} + b

x^{2} dy/dx + 2xy = 3ax^{2} + 0

â‡’ x dy/dx + 2y = 3ax ...........(I)

â‡’ x d^{2}y/dx^{2} + dy/dx + 2 dy/dx = 3a

â‡’ x d^{2}y/dx^{2} + 3 dy/dx = 3a

â‡’ 3a = x d^{2}y/dx^{2} + 3 dy/dx .................. (II)

From (1) and (2)

â‡’ x dy/dx + 2y = (x d^{2}y/dx^{2} + 3 dy/dx)x

â‡’ x dy/dx + 2y = x^{2} d^{2}y/dx^{2} + 3x dy/dx

â‡’ x^{2} d^{2}y/dx^{2} + 3x dy/dx - x dy/dx - 2y = 0

â‡’ x^{2} d^{2}y/dx^{2} + 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: x^{2} = 4ay (One constant: a)

Differential equation: x. dy/dx = 2y

x^{2}/b^{2} + y^{2}/a^{2} = 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 d^{2}y/dx^{2} + x(dy/dx)^{2} - y dy/dx = 0

3. Eliminate the arbitrary constants using the equations obtained.

xy = ce^{x} + be^{-x} + x^{2} Constants: c,b

Differentiating both sides with respect to x:

x . dy/dx + y = ce^{x} - be^{-x} + 2xâ€¦â€¦. (1)

Differentiating both sides with respect to x again:

x d^{2}y/dx^{2} + dy/dx + dy/dx = ce^{x} + be^{-x} + 2

â‡’ x d^{2}y/dx^{2} + 2. dy/dx = ce^{x} + be^{-x} + 2 â€¦.. (2)

â‡’ x d^{2}y/dx^{2} + 2. dy/dx = xy - x^{2} + 2 [Since, xy = ce^{x} + be^{-x} + x^{2}]

Required differential equation

x d^{2}y/dx^{2} + 2. dy/dx - xy + x^{2} - 2 = 0.