The general solution to the differential equation 1. Separate the variables Rearrange the equation to group all terms on one side and all terms on the other:
−1y=2x3+Cnegative 1 over y end-fraction equals 2 x cubed plus cap C 3. Solve for To get the explicit solution, isolate . First, multiply the entire equation by -1negative 1
[ \frac{1}{y} = -2x^3 - C ]
Using the Power Rule for integration, $\int u^n du = \frac{u^{n+1}}{n+1} + C$, we increase the exponent by 1 (from -2 to -1) and divide by the new exponent.
For (y = 0): (\frac{dy}{dx} = 0 = 6x^2 \cdot 0). Correct. solve the differential equation. dy dx 6x2y2
integral of y to the negative 2 power space d y equals integral of 6 x squared space d x 3. Solve the integrals Using the power rule for integration ( Left side: Right side:
We have now successfully separated the variables. The $y$ terms are isolated on the left, and the $x$ terms are isolated on the right. We are now ready to integrate. The general solution to the differential equation 1
Now the variables are separated: the left side depends only on (y), and the right side depends only on (x).
In our case, (g(x) = 6x^2) and (h(y) = y^2). First, multiply the entire equation by -1negative 1
(\frac{1}{y} = -2x^3 + K)