Login

Solve the differential equation:

x dy/dx + y = x cosx + sinx

given that y(pie/2) =1.

$Given x \frac{ⅆ y}{ⅆ x} + y = x \cos x + \sin x \frac{ⅆ y}{ⅆ x} + \frac{y}{x} = \frac{x \cos x + \sin x}{x} This is linear differenciation equ ation of form \frac{ⅆ y}{ⅆ x} + P y = Q Here P= \frac{1}{x} and Q= \frac{x \cos x + \sin x}{x} ∴ {Integrating factor is e}^{\int P ⅆ x} = e^{\int \frac{1}{x} ⅆ x} = e^{\log x} = x ∴ Required solution is y. e^{\int P ⅆ x} = \int Q . e^{\int P ⅆ x} ⅆ x \Rightarrow y . e^{\int \frac{1}{x} ⅆ x} = \int \frac{x \cos x + \sin x}{x} . e^{\int \frac{1}{x} ⅆ x} ⅆ x \Rightarrow y e^{\log x} = \int \frac{x \cos x + \sin x}{x} . e^{\log x} ⅆ x \Rightarrow y . x = \int \frac{x \cos x + \sin x}{x} . x ⅆ x \Rightarrow x y = \int x \cos x + \sin x ⅆ x \Rightarrow x y = \int x \cos x ⅆ x + \int \sin x ⅆ x \Rightarrow x y = x \sin x - \int \sin x ⅆ x + \int \sin x ⅆ x \Rightarrow x y = x \sin x \Rightarrow y = \sin x Given y(\frac{π}{2}) = 1 \Rightarrow \sin \frac{π}{2} = 1$

77

View Full Answer

Copyright © 2024 Aakash EduTech Pvt. Ltd. All rights reserved.

Aakash EduTech Pvt. Ltd.

What are you looking for?