To find the general solution of the differential equation given by y dx + 6xy^8 dy = 0, we will follow a systematic approach.
First, let’s rewrite the equation in a more manageable form:
y dx + 6xy^8 dy = 0
This can be rearranged into the standard differential form:
y dx = -6xy^8 dy
Now, we can separate the variables by dividing both sides by xy^8
frac{dx}{x} = -6y^7 dy
Next, we will integrate both sides. The left side is integrated with respect to x and the right side with respect to y:
∫ (1/x) dx = ∫ -6y^7 dy
Calculating these integrals gives us:
ln|x| = - rac{6}{8} y^8 + C
which simplifies to:
ln|x| = - rac{3}{4} y^8 + C
To express this in terms of x, we can exponentiate both sides:
|x| = e^{C} e^{- rac{3}{4} y^8}
Let K = e^{C}, a constant, so we have:
|x| = K e^{- rac{3}{4} y^8}
Therefore, we can write the general solution of the differential equation as:
x = K e^{- rac{3}{4} y^8}
Where K is an arbitrary constant. This expression represents the relationship between x and y for the given differential equation.