What is the solution for (dy)/(dx)=sqrt(y),y(1)=1 ?

The solution for (dy)/(dx)=sqrt(y),y(1)=1 is y=(x^2)/4+x/2+1/4

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(dy)/(dx)=sqrt(y),y(1)=1

\frac{d y}{d x} = y, y (1) = 1

y = \frac{x ^{2}}{4} + \frac{x}{2} + \frac{1}{4}

Solution steps

\frac{d y}{d x} = y

Solve separable ODE:

y = \frac{x ^{2}}{4} + \frac{x}{2} + \frac{1}{4}

y = \frac{x ^{2}}{4} + \frac{x}{2} + \frac{1}{4}

y = e^{\frac{3}{2 x}} \frac{d y}{d x}

e^{x} (x + 1) y^{2} d x + 2 x e^{x} y^{1} d y = 0

x^{2} \frac{d y}{d x} = (2 y)

3 (2 + x^{2}) \frac{d y}{d x} = 2 x y (y^{3} - 1)

y^{^{'}} = 1 + 0.001 y^{2}

What is the solution for (dy)/(dx)=sqrt(y),y(1)=1 ?
The solution for (dy)/(dx)=sqrt(y),y(1)=1 is y=(x^2)/4+x/2+1/4