General And Particular Solutions Of A Differential Equation

f(x)%3D%5Cfrac%7Be%5E%7B2%20x%7D-1%7D%7Be%5E%7B2%20x%7D%2B1%7D

f(x)%3D%5Cfrac%7B%5Cleft%5Be%5E%7B2%20x%7D%2B1%5Cright%5D%5Cleft%5B2%20e%5E%7B2%20x%7D%5Cright%5D-%5Cleft%5Be%5E%7B2%20x%7D-1%5Cright%5D%5Cleft%5B2%20e%5E%7B2%20x%7D%5Cright%5D%7D%7B%5Cleft%5Be%5E%7B2%20x%7D%2B1%5Cright%5D%5E%7B2%7D%7D

f%20%5E%7B1%7D(%20x%20)%3D%5Cfrac%7B2%20e%20%5E%7B2%20x%20%7D%5Cleft%5B%20e%20%5E%7B2%20x%20%7D%2B1-%20e%20%5E%7B2%20x%20%7D%2B1%5Cright%5D%7D%7B%5Cleft%5B%20e%20%5E%7B2%20x%20%7D%2B1%5Cright%5D%5E%7B2%7D%7D

f%20%5E%7B1%7D(%20x%20)%3D%5Cfrac%7B4%20e%20%5E%7B2%20x%20%7D%7D%7B%5Cleft%5B%20e%20%5E%7B2%20x%20%7D%2B1%5Cright%5D%5E%7B2%7D%7D

Now,

%5Cleft%5B%20e%20%5E%7B2%20x%20%7D%2B1%5Cright%5D%5E%7B2%7D%3E0

and

e%20%5E%7B2%20x%20%7D%3E0

%5Ctherefore%20f%20%5E%7B1%7D(%20x%20)%3E0

%5Ctherefore%20f(x)

is increasing function.

Given Equation is

%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%2B%5Cfrac%7B2%20y%20x%7D%7B1%2Bx%5E%7B2%7D%7D%3D%5Cfrac%7B1%7D%7B%5Cleft(1%2Bx%5E%7B2%7D%5Cright)%5E%7B2%7D%7D

It is comparing with linear differential equation

%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%2Bp%20y%3DQ

, we get

p%20%3D%5Cfrac%7B2%20x%20%7D%7B1%2B%20x%20%5E%7B2%7D%7D

and

Q%20%3D%5Cfrac%7B1%7D%7B%5Cleft(1%2B%20x%20%5E%7B2%7D%5Cright)%5E%7B2%7D%7D

Now, IF

%3De%5E%7BP%20d%20x%7D%3De%5E%7B%5Cfrac%7B2%20x%7D%7B1%2Bx%5E%7B2%7D%7D%20d%20x%7D

e%5E%7B%5Cleft(%5Clog%201%2Bx%5E%7B2%7D%5Cright)%7D%3D1%2Bx%5E%7B2%7D

Solution of differential equation is

y%5Cleft(1%2Bx%5E%7B2%7D%5Cright)%3D%5Cint%20%5Cfrac%7B1%7D%7B%5Cleft(1%2Bx%5E%7B2%7D%5Cright)%5E%7B2%7D%7D%5Cleft(1%2Bx%5E%7B2%7D%5Cright)%20d%20x%2Bc

%5CRightarrow%20y%20%5Cleft(1%2B%20x%20%5E%7B2%7D%5Cright)%3D%5Cint%20%5Cfrac%7B1%7D%7B%5Cleft(1%2B%20x%20%5E%7B2%7D%5Cright)%7D%20d%20x%20%2B%20c

%5CRightarrow%20y%20%5Cleft(1%2B%20x%20%5E%7B2%7D%5Cright)%3D%5Ctan%20%5E%7B-1%7D%20x%20%2B%20c

%5CRightarrow%20y%20%3D%5Cfrac%7B%5Ctan%20%5E%7B-1%7D%20x%20%7D%7B1%2B%20x%20%5E%7B2%7D%7D%2B%20c

Chapter Questions
4 MCQs