Solution Of Differential Equations

This is a first-order linear differential equation. The standard form is:

%5Cfrac%7Bdy%7D%7Bdx%7D%20%2B%20P(x)%20y%20%3D%20Q(x)

Here,

P(x)%20%3D%20%5Ccot%20x

and

Q(x)%20%3D%204x%20%5Ccsc%20x

. The integrating factor

%5Cmu(x)

is given by:

%5Cmu(x)%20%3D%20e%5E%7B%5Cint%20P(x)%20%5C%2C%20dx%7D%20%3D%20e%5E%7B%5Cint%20%5Ccot%20x%20%5C%2C%20dx%7D%20%3D%20e%5E%7B%5Cln%20%5Csin%20x%7D%20%3D%20%5Csin%20x

Multiply both sides of the differential equation by

%5Cmu(x)%20%3D%20%5Csin%20x

%5Csin%20x%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%2B%20y%20%5Csin%20x%20%5Ccot%20x%20%3D%204x

This simplifies to:

%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20y%20%5Csin%20x%20%5Cright)%20%3D%204x

Now, integrate both sides:

%5Cint%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20y%20%5Csin%20x%20%5Cright)%20%5C%2C%20dx%20%3D%20%5Cint%204x%20%5C%2C%20dx

y%20%5Csin%20x%20%3D%202x%5E2%20%2B%20C

Now, use the initial condition

y%20%3D%200

when

x%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D

0%20%5Ccdot%20%5Csin%20%5Cleft(%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cright)%20%3D%202%20%5Cleft(%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cright)%5E2%20%2B%20C

0%20%3D%202%20%5Ctimes%20%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D%20%2B%20C%20%5C(%5CRightarrow%5C)%20C%20%3D%20-%5Cfrac%7B%5Cpi%5E2%7D%7B2%7D

Thus, the particular solution is:

y%20%5Csin%20x%20%3D%202x%5E2%20-%20%5Cfrac%7B%5Cpi%5E2%7D%7B2%7D

y%20%3D%20%5Cfrac%7B2x%5E2%20-%20%5Cfrac%7B%5Cpi%5E2%7D%7B2%7D%7D%7B%5Csin%20x%7D

Final Answer: The particular solution is:

y%20%3D%20%5Cfrac%7B2x%5E2%20-%20%5Cfrac%7B%5Cpi%5E2%7D%7B2%7D%7D%7B%5Csin%20x%7D

The given differential equation is:

%5Cfrac%7Bdy%7D%7Bdx%7D%20%2B%20y%20%5Ccot%20x%20%3D%202x%20%2B%20x%5E2%20%5Ccot%20x

This is a linear first-order differential equation of the form:

%5Cfrac%7Bdy%7D%7Bdx%7D%20%2B%20P(x)y%20%3D%20Q(x)

Where

P(x)%20%3D%20%5Ccot%20x

and

Q(x)%20%3D%202x%20%2B%20x%5E2%20%5Ccot%20x

. The integrating factor

%5Cmu(x)

is given by:

%5Cmu(x)%20%3D%20e%5E%7B%5Cint%20P(x)%20%5C%2C%20dx%7D%20%3D%20e%5E%7B%5Cint%20%5Ccot%20x%20%5C%2C%20dx%7D%20%3D%20e%5E%7B%5Cln%20%5Csin%20x%7D%20%3D%20%5Csin%20x

Multiply both sides of the differential equation by

%5Cmu(x)%20%3D%20%5Csin%20x

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

This simplifies to:

%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20y%20%5Csin%20x%20%5Cright)%20%3D%202x%20%5Csin%20x%20%2B%20x%5E2%20%5Ccos%20x

Now, integrate both sides:

%5Cint%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cleft(%20y%20%5Csin%20x%20%5Cright)%20%5C%2C%20dx%20%3D%20%5Cint%20%5Cleft(%202x%20%5Csin%20x%20%2B%20x%5E2%20%5Ccos%20x%20%5Cright)%20%5C%2C%20dx

The left side is:

y%20%5Csin%20x

To integrate the right-hand side, break it into two integrals:

%5Cint%202x%20%5Csin%20x%20%5C%2C%20dx%20%5Ctext%7Band%7D%20%5Cint%20x%5E2%20%5Ccos%20x%20%5C%2C%20dx

Use integration by parts for both terms. For

%5Cint%202x%20%5Csin%20x%20%5C%2C%20dx

, we get:

%5Cint%202x%20%5Csin%20x%20%5C%2C%20dx%20%3D%20-2x%20%5Ccos%20x%20%2B%202%20%5Csin%20x

For

%5Cint%20x%5E2%20%5Ccos%20x%20%5C%2C%20dx

, use integration by parts again:

%5Cint%20x%5E2%20%5Ccos%20x%20%5C%2C%20dx%20%3D%20x%5E2%20%5Csin%20x%20-%202x%20%5Ccos%20x%20%2B%202%20%5Csin%20x

Thus, the right-hand side becomes:

-2x%20%5Ccos%20x%20%2B%202%20%5Csin%20x%20%2B%20x%5E2%20%5Csin%20x%20-%202x%20%5Ccos%20x%20%2B%202%20%5Csin%20x%20%3D%20-4x%20%5Ccos%20x%20%2B%20x%5E2%20%5Csin%20x%20%2B%204%20%5Csin%20x

Thus, the general solution is:

y%20%5Csin%20x%20%3D%20-4x%20%5Ccos%20x%20%2B%20x%5E2%20%5Csin%20x%20%2B%204%20%5Csin%20x%20%2B%20C

Now, substitute

x%20%3D%20%5Cfrac%7B%5Cpi%7D%7B2%7D

and

y%20%3D%200

0%20%5Ccdot%201%20%3D%20-4%20%5Ccdot%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Ccdot%200%20%2B%20%5Cleft(%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cright)%5E2%20%5Ccdot%201%20%2B%204%20%5Ccdot%201%20%2B%20C

Solving for

C

C%20%3D%20-4%20%2B%20%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D%20%2B%204%20%3D%20%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D

Thus, the particular solution is:

y%20%5Csin%20x%20%3D%20-4x%20%5Ccos%20x%20%2B%20x%5E2%20%5Csin%20x%20%2B%204%20%5Csin%20x%20%2B%20%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D

y%20%3D%20%5Cfrac%7B-4x%20%5Ccos%20x%20%2B%20x%5E2%20%5Csin%20x%20%2B%204%20%5Csin%20x%20%2B%20%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D%7D%7B%5Csin%20x%7D

Final Answer:

%5Cboxed%7By%20%3D%20%5Cfrac%7B-4x%20%5Ccos%20x%20%2B%20x%5E2%20%5Csin%20x%20%2B%204%20%5Csin%20x%20%2B%20%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D%7D%7B%5Csin%20x%7D%7D

About Solution Of Differential Equations - UP-BOARD-XII

Solution Of Differential Equations is a vital chapter for UP-BOARD-XII aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

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Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

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Review the topic breakdown to see which sub-topics within Solution Of Differential Equations carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

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About Solution Of Differential Equations - UP-BOARD-XII

Frequently Asked Questions

Why focus on Solution Of Differential Equations PYQs?

How to best use this analysis?

Solution Of Differential Equations

High-Yield Trend

Chapter Questions 6 MCQs

Official Solution

Official Solution

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Official Solution

About Solution Of Differential Equations - UP-BOARD-XII

Frequently Asked Questions

Why focus on Solution Of Differential Equations PYQs?

How to best use this analysis?

Chapter Questions
6 MCQs