Please help me with this diffrential equations question

Question

Find an equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is 15xy.

Answer 1

The slope of the curve is 15xy.

Therefore:

dy/dx = 15xy

dy/y = (15x)dx

We can then integrate both sides giving us

ln(y) = (15x^2)/2 + C

the curve passes through (0,1) so we substitute x=0 and y=1 to solve for C

ln(1) = 0/2 + C

0 = 0 + C

C = 0

We now substitute C back in and isolate y

ln(y) = (15x^2)/2

y = e^{(15x^2) / 2}

The answer is y = e^{(15x^2) / 2}