Solve The - Differential Equation. Dy Dx 6x2y2

We know $y = \frac{1}{C - 2x^3}$. Therefore, $y^2 = \frac{1}{(C - 2x^3)^2}$.

−y-1=6x33+Cnegative y to the negative 1 power equals the fraction with numerator 6 x cubed and denominator 3 end-fraction plus cap C Simplify the right side: solve the differential equation. dy dx 6x2y2

Using the Power Rule again (increasing the exponent from 2 to 3 and dividing by 3): $$ 6 \left( \frac{x^3}{3} \right) $$ We know $y = \frac{1}{C - 2x^3}$

If you encounter a similar problem, follow these steps: Simple enough

[ \frac{dy}{dx} = -1 \cdot (K - 2x^3)^{-2} \cdot (-6x^2) = \frac{6x^2}{(K - 2x^3)^2} ]

That’s the general solution. Simple enough.

If (y = 0) for all (x), then ( \frac{dy}{dx} = 0) and (6x^2 y^2 = 6x^2 \cdot 0 = 0). So (y = 0) is also a solution. It’s not covered by the formula above (which would require (C \to \infty) to get zero), so we note it as a singular solution .