Differential Equation

A differential equations are a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equations. The order of the highest derivative included in a the differential equation defines the order of the differential equation.(Source : WIKIPEDIA)

Sample Differential Equations practice test :

Example 1:
     Solve the given differential equations for dy + 9x dx = 0
Solution:
We simply need to subtract
     9x dx
from both sides, then insert integral signs and integrate:
     dy = -9x dx
     dy = - 9x dx
     y = -`(9)/(2)` x² + K
NOTE 1: We could have written it in a more familiar way as: `(dy)/(dx) ` + 9x = 0
     Then `(dy)/(dx)` = -9x
     So y = -`(9)/(2)` x² + K
NOTE 2: ∫dy means ∫1dy. We also have:
     ∫dt = t    ∫dθ = θ   ∫da = a  and so on.

Example 2:

Show that y = c₁ sin 3x + 2 cos 3x is a general solution for the differential equation `(d^2y)/(dx^2 )` + 9y = 0

Solution:
We need to find the second derivative of y:
      y = c₁ sin 3x + 2 cos 3x
First derivative:
`(dy)/(dx)` = 3c₁ cos3x – 6sin3x
Second derivative:
`(d^2y)/(dx^2) ` = -9c₁sin3x – 18cos3x
Now for the check step:
LHS = `(d^2y)/(dx^2)` + 9y
        = [-9c₁sin3x – 18cos3x] + 9[c₁sin3x + 2cos3x]
        = 0
        = RHS

Example 3:

Solve  the given differential equation 2 y ' = sin(2x)

Solution:

Write the differential equation of the form y ’ = f(x).

     y ' = (1/2) sin(2x)

Integrate on both sides

    ∫y ' dx = ∫ (1/2) sin(2x) dx

Let u = 2x so that du = 2dx the right side becomes

     y = ∫ (1/4) sin(u) du

Which gives

     y = (-1/4) cos(u) = (-1/4) cos (2x)

Example 4:
Solve the given differential equation y 'e ^-x + e ^2x = 0

Multiply all terms of the equation by e x and write the differential equation of the form y ' = f(x).

     y ' = - e ^3x

Integrate both sides of the equation

     ∫y ' dx = ∫ - e ^3x dx
Let u = 3x so that du = 3 dx,
     y = ∫ (-1/3) e ^u du

Which gives.

     y = (-1/3) e ^u = (-1/3) e ^3x

Practice Differential Equations test problems:

Practice test problem:- 1
Solve the given differential equations for dy + 5x dx = 0
Answer: -`(5)/(2)` x² + K
Practice test problem:- 2
Show that y = c₁ sin 5x + 2 cos 5x is a general solution for the differential equation `(d^2y)/(dx^2 )` + 25y = 0
Practice test problem:- 3
Solve  the given differential equation 3 y ' = sin(3x)
Answer: y= (-1/9) cos (3x)
Practice test problem:- 4
Solve the given differential equation y ' e ^x = e ^3x
Answer: y =(1 / 2) e ^2x + C