How to Solve Differential Equation with Example

Differential Equation

– are equation that contain differential coefficients.

Example:

– classified according to the highest derivative that occurs in Them the differential equation dy/dx = 12x is a first order differential equation d^2y/dx^2 + 4dy/dx – 3y = 0 is a second order differential equation. A solution to a differential equation that contains one or more arbitrary constant of integration is called general solution. When additional information is Given so that these constants may be calculated the *particular solution* of differential equation is obtained.

Variable Separable

A differential equations can be of type dy/dx = f(x)solved by direct integration by writing it in the form dy = f(x) dx

Example:

Solve the differential equation dy/dx = 2x + Sin 3x

Solution: dy = (2x + Sin 3x) dx

y = x2 – 1/3 Cos 3x + c * general solution*

Differential equation of type dy/dx = f(y) can be solved by direct integrating by writing it in the form.

dx = dy/f(x)

Example:

Solve the equation (y^2 – 1) dy/dx = 3y Given that y = 1 when x = 13/6

Differential Equation of type

dy/dx = f(x) g(y) can be solve by Direct Integration by writing it in the form dy/g(y) = f(x)dx

Differential Equation of type dq/dt = KQ the general solution of an equation of the form dq/dt = KQ is Q = Ce^kt . where C is constant

Example:

solve the equation dy/dx = 3y

here we have Q = y dQ = dy then

t = x k = 3 y = Ce^3x

Example:

Obtain the differential equation of the family of straight lines with slope and y intercept equal

Solution:

The standard equation of the line in slope – intercept form is y = mx + b.

Since the slope and y intercept are equal m = b = c then y = cx + c

isolating constant and differentiate