Q. 234.8( 4 Votes )

Solve the differential equation: ; given that y = 1 when x = 1.

Answer :

Given,

To Find: Find the solution when y = 1 and x = 1


Explanation:


It can be written as:



…(i)


Let, F(x, y) =


Now, put x = λx and y = λy in F(x, y)



Taking λ as common from both numerator and denominator



We know, If the degree of the λ in function F(x, y) then it is said Homogenous function


So, the given differential equation is Homogenous differential equation.


Put y = vx in equation(i)


Differentiation y w.r.t x , we get


…(iii)


Now, Compare the equation (i) and (iii), we get



Taking x2 as common from both numerator and denominator







Integrating both sides, we get



Let v2 + 1 = t


On differentiating we get,


2v dv = dt, So



log t = - log x + log C


Putting the value of t , we get


Log|v2 + 1| + log|x| = log|c|


log|x(v2 + 1)| = log|C|


Now, putting back the value of v = y/x ,




- - - (iv)


When, x = 1 and y = 1


(1)2 + 1 = C(1)


C = 2


Put the value of C in equation (iv)


y2 + 1 = x(2)


y2 = 2x - 1


Hence, y2 = 2x - 1 is the solution of given differential equation.


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