Part (b)
We use the result of part (a) and plug in (x,y) = (0,0). This is directly from the initial condition y(0) = 0.
[tex]\arcsin(4y) = x^2 + C\\\\\arcsin(4*0) = (0)^2 + C\\\\\arcsin(0) = C\\\\0 = C\\\\C = 0\\\\[/tex]
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This means,
[tex]\arcsin(4y) = x^2 + C\\\\\arcsin(4y) = x^2 + 0\\\\\arcsin(4y) = x^2\\\\4y = \sin(x^2)\\\\y = \frac{1}{4}\sin(x^2)\\\\[/tex]
is the solution with the initial condition y(0) = 0.
