Order And Degree Of Differential Equation

We are given the differential equation:

%5Cfrac%7Bdx%7D%7Bdy%7D%20%3D%20%5Cfrac%7B%5Csin%20y(1%20%2B%20y%20%5Ccot%20y)%7D%7Bx%20%5Clog(x%5E2%20e)%7D

We also know the initial condition

y(1)%20%3D%200

.
Step 1: Simplify the equation.
First, simplify the expression for

%5Clog(x%5E2%20e)

%5Clog(x%5E2%20e)%20%3D%20%5Clog(x%5E2)%20%2B%20%5Clog(e)%20%3D%202%20%5Clog%20x%20%2B%201

Substitute this into the differential equation:

%5Cfrac%7Bdx%7D%7Bdy%7D%20%3D%20%5Cfrac%7B%5Csin%20y%20(1%20%2B%20y%20%5Ccot%20y)%7D%7Bx%20(2%20%5Clog%20x%20%2B%201)%7D

Step 2: Solve the differential equation.
We use the method of separation of variables. First, multiply both sides by

x%20(2%20%5Clog%20x%20%2B%201)

to separate the variables:

x%20(2%20%5Clog%20x%20%2B%201)%20%5C%2C%20dx%20%3D%20%5Csin%20y%20(1%20%2B%20y%20%5Ccot%20y)%20%5C%2C%20dy

Now, integrate both sides:

%5Cint%20x%20(2%20%5Clog%20x%20%2B%201)%20%5C%2C%20dx%20%3D%20%5Cint%20%5Csin%20y%20(1%20%2B%20y%20%5Ccot%20y)%20%5C%2C%20dy

The integration on the left-hand side leads to:

%5Cint%20x%20(2%20%5Clog%20x%20%2B%201)%20%5C%2C%20dx%20%3D%20x%5E2%20%5Clog%20x

For the right-hand side, solving the integral gives:

%5Cint%20%5Csin%20y%20(1%20%2B%20y%20%5Ccot%20y)%20%5C%2C%20dy%20%3D%20y%20%5Csin%20y

Step 3: Apply the initial condition.
Using the initial condition

y(1)%20%3D%200

, substitute into the equation:

y%20%5Csin%20y%20%3D%20x%5E2%20%5Clog%20x

This is the required particular solution. Final Answer:

%5Cboxed%7By%20%5Csin%20y%20%3D%20x%5E2%20%5Clog%20x%7D

Chapter Questions
1 MCQs