Q26 of 45 Page 1

Show that the differential equation 2yex/y dx + (y - 2xex/y)dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.

Given differential equation is

…(i)


Since the variable is in the form we will take instead of


Firstly, we will find the










Now, we have to find F(λx, λy)




⇒ F(λx, λy) = F(x, y)


⇒ F(λx, λy) = λ0 F(x, y)


Thus, F(x, y) is a homogeneous function of degree zero.


∴ given differential equation is homogeneous differential equation.


Now, we will solve the differential equation by putting x = vy


…(ii)


Putting x = vy


Differentiate with respect to y, we get





Now, putting the values of and x in eq. (ii), we get






Integrating both the sides, we get



⇒ 2ev = - log |y| + C



[using x = vy]


…(iii)


It is given that x = 0 and y = 1



⇒ 2 × 1 + 0 = C [∵ log |1| = 0] ⇒ C = 2


Putting the value of C = 2 in eq. (iii), we get



is the particular solution of differential equation


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