Differential Equations

Given,

%5Cdisplaystyle%5Clim_%7Bt%5Cto%20x%7D%20%5Cfrac%7Bt%5E%7B2%7Df%5Cleft(x%5Cright)-x%5E%7B2%7Df%5Cleft(t%5Cright)%7D%7Bt-x%7D%3D1

Apply L' Hospital rule

%5Cdisplaystyle%5Clim_%7Bt%5Cto%20x%7D%20%5Cfrac%7B2t%5C%2Cf%20%5Cleft(x%5Cright)-x%5E%7B2%7Df'%5Cleft(t%5Cright)%7D%7B1%7D%3D1

%5CRightarrow%202x%20f%20%5Cleft(x%5Cright)-x%5E%7B2%7Df'%5Cleft(x%5Cright)%3D1

%5CRightarrow%20x%5E%7B2%7Df'%20%5Cleft(x%5Cright)-2x%20f%5Cleft(x%5Cright)%3D-1

%5CRightarrow%20f'%5Cleft(x%5Cright)-%5Cfrac%7B2%7D%7Bx%7D%20f%20%5Cleft(x%5Cright)%3D-%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D

It is a linear differential equation

%5Ctherefore%20IF%20%3De%5E%7B%5Cint%20%5Cfrac%7B2%7D%7Bx%7Ddx%7D

%3De%5E%7B-2%5C%2Clog%5C%2Cx%7D

%3De%5E%7Blog%20%5Cleft(%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%5Cright)%7D

%3D%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D

%5Ctherefore

Solution is

f%20%5Cleft(x%5Cright)%5Ctimes%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%3D%5Cint-%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%5Ctimes%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%20dx%2BC

f%20%5Cleft(x%5Cright)%5Ctimes%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%3D%5Cfrac%7B1%7D%7B3x%5E%7B3%7D%7D%2BC

%5CRightarrow%20f%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B3x%7D%2BCx%5E%7B2%7D

f%5Cleft(1%5Cright)%3D1

%5CRightarrow%201%3D%5Cfrac%7B1%7D%7B3%7D%2BC

%5CRightarrow%20C%3D%5Cfrac%7B2%7D%7B3%7D

%5Ctherefore%20f%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B3x%7D%2B%5Cfrac%7B2%7D%7B3%7Dx%5E%7B2%7D

Given,

x%5C%2C%20cos%5C%2C%20x%20%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%2By%20%5Cleft(x%5C%2Csin%5C%2Cx%2Bcos%5C%2Cx%5Cright)%3D1

%5CRightarrow%20%5Cfrac%7Bdy%7D%7Bdx%7D%2B%5Cfrac%7By%20%5Cleft(x%5C%2Csin%5C%2Cx%2Bcos%5C%2Cx%5Cright)%7D%7Bx%5C%2C%20cos%5C%2Cx%7D

%3D%5Cfrac%7B1%7D%7Bx%5C%2C%20cos%5C%2Cx%7D

On comparing with

%5Cfrac%7Bdy%7D%7Bdx%7D%2Bpy%3DQ

, we get

P%3D%5Cfrac%7Bx%5C%2Csin%5C%2Cx%2Bcos%5C%2Cx%7D%7Bx%5C%2Ccos%5C%2Cx%7D

%3Dtan%5C%2Cx%2B%5Cfrac%7B1%7D%7Bx%7D

%5Ctherefore%20IF%3De%5E%7B%5Cint%20p%5C%2Cdx%20%7D

%3De%5E%7B%5Cint%5Cleft(tan%5C%2Cx%2B%5Cfrac%7B1%7D%7Bx%7D%5Cright)dx%7D

%3De%5E%7Blog%5C%2Csec%5C%2Cx%2Blog%5C%2Cx%7D

%3De%5E%7Blog%5C%2Cx%5C%2Csec%5C%2Cx%7D%3Dx%5C%2Csec%5C%2Cx

%5Ctherefore

Solution is

y%5Ctimes%20x%5C%2Csec%5C%2Cx%3D%5Cint%20%5Cfrac%7Bx%5C%2Csec%5C%2Cx%7D%7Bx%5C%2Ccos%5C%2Cx%7D%20dx

%3D%5Cint%20sec%5E%7B2%7Dx%5C%2Cdx

%5CRightarrow%20xy%5C%2C%20sec%5C%2C%20x%20%3Dtan%5C%2C%20x%2Bc

About Differential Equations - AMUEEE

Differential Equations is a vital chapter for AMUEEE 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.

Frequently Asked Questions

Why focus on Differential Equations PYQs?

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.

How to best use this analysis?

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

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

About Differential Equations - AMUEEE

Frequently Asked Questions

Why focus on Differential Equations PYQs?

How to best use this analysis?

Differential Equations

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

About Differential Equations - AMUEEE

Frequently Asked Questions

Why focus on Differential Equations PYQs?

How to best use this analysis?

Chapter Questions
3 MCQs