Solve The Differential Equation. Dy Dx 6x2y2
Let $u = C - 2x^3$. Then $y = u^{-1}$.
$$ \frac{y^{-1}}{-1} = -\frac{1}{y} $$ Now we integrate the right side with respect to $x$: $$ \int 6x^2 , dx $$ solve the differential equation. dy dx 6x2y2
Depending on the textbook or context, you might see the constant handled differently. Sometimes it is cleaner to define a new constant $A = -C$. Let's look at the result if we clean up the negative sign in the denominator: Let $u = C - 2x^3$
Because we can separate the equation into an $x$-side and a $y$-side, this is known as a . The strategy for solving separable equations is straightforward: separate the variables, integrate both sides, and solve for $y$. Step 1: Separation of Variables The goal of this step is to rearrange the equation so that all terms involving $y$ are on the side with $dy$, and all terms involving $x$ are on the side with $dx$. Sometimes it is cleaner to define a new constant $A = -C$
Substitute $y^2$ back into the original right-hand side expression $6x^2y^2$: $$ 6x^2y^2 = 6x^2 \left( \frac{1}{(C - 2x^3)^2} \right) = \frac{6x^2}{(C-2x^3)^2} $$