What is the particular solution of the differential equation $dy/dx = xy^(1/2)$ with $y(0)=0$ ?

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Answer 1 · 2018-02-04T00:32:05

$y = x^4/16$

We have:

$dy/dx = xy^(1/2)$ with $y(0)=0$

This is a First Order Separable ODE, so we can can write:

$y^(-1/2) \ dy/dx = x$

Now, we separate the variables to get

$int \ y^(-1/2) \ dy = int \ x \ dx$

Which consists of standard integrals, so we can integrate:

$y^(1/2)/(1/2) = x^2/2 + C$

Applying the initial condition, we have:

$0 = 0 + C => C = 0$

Thus, we have:

$y^(1/2)/(1/2) = x^2/2$
$:. y^(1/2) = x^2 /4$
$:. y = x^4/16$

What is the particular solution of the differential equation dy/dx = xy^(1/2) dy/dx = xy^(1/2) with y(0)=0y(0)=0?