Practice Problems on Differential Equations

A differential equation is an equation that has a function, such as F(x), and one or even more derivatives, such as dy/dx. The physical quantities are commonly represented by the functions of a differential equation, while the rate of change of the physical quantities is given by the derivatives. A differential equation is a mathematical expression that describes the connection between a function and its derivatives. When we examine y as a function of x, a differential equation is the one that incorporates derivatives of y with respect to x (or the differentials of y and x) with or without variables x and y.

Order of differential equations

It represents the highest order of the derivative. Here are some of the examples:

1. (dy/dx) = sec x
2. (d2y/dx2) + p2y = 0
3. (2d2y/dt2) + (5d2x/dt2) = 8x
4. (d3y/dx3) + 7x(dy/dx) – 9xy = 10
5. (rdr/dθ) + tanθ = 10

• In the first case, the order is 1
• In the second case, the order is 2 as the equation has second-order derivative d2 y/ dx2
• In the third case, the order is 2 as the equation has second-order derivative d2 y/ dx2
• In the fourth case, the order is 3 as the equation has second-order derivative d3 y/ dx3
• In the fifth case, the order is 1

Degree of Differential Equations

• The highest-order derivatives power is called the degree of the differential equation.
• The differential equation may be in the polynomial form in that the highest order of differential power is called a degree.
• Choose the positive integer and find the highest degree:

( d5ydx5 )4  + 7 ( dydx )8  + 9y = 7 sec7x

, where dx5  is the highest order of the equation, so the highest degree is 4

Types of Differential Equation

There are two types of differential equations. They are:

1. Ordinary differential equation
2. Partial differential equation

Ordinary Differential Equations

• ODE – A differential equation comprising one or even more functions from one independent variable with their derivatives is known as an ordinary differential equation (ODE).

For example here are some derivatives y’+y”+y”’+y””+……… yn concerning x

Example: (d2y/dx2) + (dy/dx) = 3y tan x

Partial Differential Equations

An equation that involves more than one function and more than one independent variable is called Partial differential equation(PDE).

Here are a few examples:

• 5𝛿u/ 𝛿x + 7𝛿u/𝛿y = 0,
• 21𝛿2u/𝛿x2 + 8𝛿2u/𝛿y2 = 0

Practice Problems on Differential Equations

• Show that the differential equation (5x- 8y).dy/dx = (4x+2y) is a homogeneous differential equation.

Solution:

(5x – 8y).dy/dx = (4x + 2y) is the given differential equation

To prove that the above differential equation is homogeneous, let us substitute x =δ x and y =δ y.

Here we have F(x, y) = (4x+2y)/(5x−8y)

F(δx, δy) = ( δ4x+δ2y)/( δ5x− δ8y) 

F(δx, δy) =  f(x, y)

Hence, it proves that the given equation is a homogeneous differential equation.

1. Find the solution of the homogeneous differential equation x sin(y/x).dy/dx = y sin(y/x) + x.

Solution:

The given differential equation is x sin(y/x).dy/dx = y sin(y/x) + x

dy/dx = {y sin(y/x) + x } / xsin(y/x)

dy/dx = {x((y/x).sin(y/x)+1)} / x sin(y/x)

dy/dx = ((y/x).sin(y/x)+1 /  sin(y/x)

Here let us replace y/x =v in the above equation

dy/dx = (v sinv+1) / sinv 

Here write y/x = v in the form of y = vx

On differentiating y = vx on both sides of the equation we obtain 

dy/dx = v + x.dv/dx, which is substituted in the above equation

v + x.dv/dx =( v sinv + 1)/ sinv     

x.dv/dx = (v sinv+ 1 / sin v) – v

Here we split the variables on both sides

x.dv/dx = 1/sinv

sinv.dv = dx/x

On integrating both equation we get the below equation

∫sinv.dv=∫dx/x

-cosv = Logx + C

Here we switch back y/x = v

-cos y/x = Logx + C

Therefore, the solution of the homogeneous differential equation is – cos y/x = Logx + C

Conclusion:

The connection between both the variables, x, and y, that are produced after eliminating the derivatives (i.e., integration) or where the connection includes an arbitrary constant to signify the order of an equation is the basic solution of a differential equation. An arbitrary constant appears in the solution of first-order differential equations, while two arbitrary constants appear in the solutions of second-order differential equations. The solution of the differential equations is derived by giving specific values to the arbitrary constant.