. Each curve represents a potential path based on specific initial conditions. Note the vertical asymptotes where the denominator equals zero. ✅ Final Answer The general solution to the differential equation
Now we put the results of both integrals back together. Usually, we combine the constants of integration from both sides into a single constant $C$ on the right side.
Now the variables are separated: the left side depends only on (y), and the right side depends only on (x). solve the differential equation. dy dx 6x2y2
$$ \frac{dy}{dx} = 6x^2y^2 $$
If you encounter a similar problem, follow these steps: ✅ Final Answer The general solution to the
This is a . It is "first-order" because the highest derivative present is the first derivative ($dy/dx$). The most critical observation here is the structure of the right-hand side. Notice that the term $6x^2y^2$ is a product of a function of $x$ (specifically $6x^2$) and a function of $y$ (specifically $y^2$).
$$ \frac{1}{y^2} \frac{dy}{dx} = 6x^2 $$ $$ \frac{dy}{dx} = 6x^2y^2 $$ If you encounter
We will walk through the method of separation of variables, perform the integration carefully, and discuss the significance of the constant of integration, including the special case of the trivial solution.
We can simplify the minus signs. Let (K = -C), where (K) is also an arbitrary constant. Then: