Find the equation of the solution to dydx=x6y through the point (x, y)=(1,3).

Separate the x's and y's.

dy/y = x^7 dx

Integrate both sides.

ln(abs(y)) = (x^8)/8 + C

To cancel the natural root, make both sides the power to e.

e^ln(abs(y)) = e^((x^8)/8 + C)

abs(y) = e^C * e^((x^8)/8)

y = + or - [e^C * e^((x^8)/8)

Now just bundle the + or - e^C into a single constant. We will call it A.

y = Ae^((x^8)/8)

Now plug in the point (1,3).

3 = Ae^((1^8)/8)

3 = Ae^(1/8)

A = 3/(e^(1/8))

So the equation is:

y = (3/(e^(1/8))*(e^((x^8)/8)