Okay, so I'm given this ODE:
(3xy-2y^2)dx - 7xydy = 0
I chose y = ux, so dy = udx + xdu
So subbing in for y I get:
(3(x^2)u - 2(x^2)(u^2))dx - 7(x^2)u(udx + xdu) = 0
3(x^2)udx - 3(x^2)(u^2)dx - 7(x^2)(u^2)dx - 7(x^3)udu = 0
3(x^2)udx - 9(x^2)(u^2)dx - 7(x^3)udu = 0
I do not know where to go from here in order to be able to integrate (i.e. separate the variables).
To integrate the given equation, we need to separate the variables and integrate each side separately. In the equation:
3(x^2)udx - 9(x^2)(u^2)dx - 7(x^3)udu = 0
First, let's factor out the common terms involving "dx" and "udu":
(3(x^2)u - 9(x^2)u^2 - 7(x^3))dx - 7(x^3)udu = 0
Now, we can divide both sides by the expression in front of "dx" and "udu":
[ (3(x^2)u - 9(x^2)u^2 - 7(x^3))dx - 7(x^3)udu ] / (3(x^2)u - 9(x^2)u^2 - 7(x^3)) = 0
The next step is to integrate both sides with respect to their respective variables.
∫ [ (3(x^2)u - 9(x^2)u^2 - 7(x^3))dx - 7(x^3)udu ] / (3(x^2)u - 9(x^2)u^2 - 7(x^3)) = ∫ 0 dx
Integrating the right side gives us:
0 + C1 = ∫ 0 dx
C1 = 0
The constant of integration (C1) is zero because we are integrating a constant value.
Next, we integrate the left side of the equation. This integral can be quite complex, so let's simplify it further before integrating.
Notice that the expression in the numerator, (3(x^2)u - 9(x^2)u^2 - 7(x^3)), can be factored:
3(x^2)u - 9(x^2)u^2 - 7(x^3) = (x^2)(3u - 7)(u - 1)
We can now rewrite the left side of the equation:
∫ [ (x^2)(3u - 7)(u - 1)dx - 7(x^3)udu ] / [(x^2)(3u - 7)(u - 1)] = 0
Let's separate the fractions:
∫ [(x^2)(3u - 7)(u - 1)dx] / [(x^2)(3u - 7)(u - 1)] - ∫ [7(x^3)udu] / [(x^2)(3u - 7)(u - 1)] = 0
Now, let's integrate both sides individually. First, we integrate the left side:
∫ [(x^2)(3u - 7)(u - 1)dx] / [(x^2)(3u - 7)(u - 1)] = ∫ dx
We can cancel out the common terms in the numerator and denominator:
∫ dx = ∫ dx
Which simplifies to:
x + C2 = x + C2
The constant of integration (C2) can be any arbitrary constant.
Now, let's integrate the right side:
-∫ [7(x^3)udu] / [(x^2)(3u - 7)(u - 1)] = 0
This integral is more complex and may require advanced techniques such as partial fractions or trigonometric substitutions, depending on the form of the denominator and numerator.
Therefore, at this stage, we have integrated both sides separately and obtained the equations:
x + C2 = x + C2
-∫ [7(x^3)udu] / [(x^2)(3u - 7)(u - 1)] = 0
To find the actual solution, we need to simplify and solve the integral involving u. Once that integral is solved, we can substitute u back into the equation and solve for the value of y.