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Vector Algebra

Question
CBSEENMA12035643
Wired Faculty App

Find the particular solution of the differential equation

2yex/y dx+ (Y-2xex/y) dy =0

Given that x=0 when y=1.

Solution
2 ye to the power of straight x over straight y end exponent dx space plus space open parentheses straight y minus 2 xe to the power of straight x over straight y end exponent close parentheses dy space space equals space 0 rightwards double arrow space dx over dy space equals space fraction numerator 2 xe to the power of begin display style straight x over straight y end style end exponent minus straight y over denominator 2 ye to the power of begin display style straight x over straight y end style end exponent end fraction Given space differential space equation space is space straight a space homogenous space differential space equation. therefore comma put space space straight x space equals space vy dx over dy space equals space straight v space plus space straight y dv over dy straight v space plus space straight y dv over dy space space space equals space fraction numerator 2 ve to the power of straight v minus 1 over denominator 2 straight e to the power of straight v end fraction rightwards double arrow space space straight y dv over dy space equals space fraction numerator 2 ve to the power of straight v minus 1 over denominator 2 straight e to the power of straight v end fraction space minus space straight v rightwards double arrow space space straight y dv over dy space equals space minus fraction numerator 1 over denominator 2 straight e to the power of straight v end fraction rightwards double arrow space space 2 straight e to the power of straight v dv space equals negative 1 over straight y dy Integrating space on space both space sides rightwards double arrow space space 2 integral straight e to the power of straight v equals negative integral 1 over straight y dy rightwards double arrow space 2 straight e to the power of straight v space space equals space log space open vertical bar straight y close vertical bar space plus space log space straight C rightwards double arrow space 2 straight e to the power of straight v space equals space log space space open vertical bar straight c over straight y close vertical bar rightwards double arrow space 2 straight e to the power of fraction numerator straight x over denominator straight y space end fraction end exponent equals space log space open vertical bar straight c over straight y close vertical bar Given space that space at space straight x equals 0 comma space straight y equals 1 2 straight e to the power of 0 space space equals space log open vertical bar straight c over 1 close vertical bar rightwards double arrow space straight C space equals space straight e squared therefore comma 2 straight e to the power of straight x over straight y end exponent space equals space log space straight e squared over straight y rightwards double arrow space space log space straight y equals negative 2 straight e to the power of straight x over straight y end exponent plus space 2 rightwards double arrow space straight y space equals space straight e to the power of 2 minus 2 straight e to the power of straight x over straight y end exponent end exponent

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = ex

Determine order and degree (if defined) of differential equations:
y'' + (y')2 + 2 y = 0

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