Q. 275.0( 1 Vote )
Solve the differential equation: x2 dy + y(x + y)dx = 0, given that y = 1 when x = 1.
Answer :
Given, A differential equation x2 dy + y(x + y)dx = 0
To Find: Find the particular solution at y = 1 and x = 1
Explanation: We have given
x2 dy + y(x + y)dx = 0
x2 dy = - y(x + y)dx
…(i)
Let F(x, y) =
To check that, given differential equation is homogenous ,
Put x = λx and y = λy in F(x, y)
Then,
Now, Taking λ2 common from both numerator and denominator
In F(x, y) If λ0, then F(x, y) is a homogenous function of degree zero.
Let’s put y = vx in equation(i)
On differentiating y , we get
…(ii)
Now, Compare the equation (i) and (ii)
Taking x2 as common from the numerator and denominator
Now, Integrating both sides
Solve dv by completing the square method
We know, , then
Now, putting
Now, Put x = 1 and y = 1
1 = 3C
Put the value of C
Hence,
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