91Ó°ÊÓ

In differential equations, 'separation of variables' is a technique used to solve ordinary differential equations. The core idea is to rearrange a given equation to isolate all terms involving the variable y on one side and all terms involving the variable x on the other side. This makes it easier to integrate each side of the equation independently.

To illustrate, let's consider the equation from our exercise: \[3 y^{2} \frac{d y}{d x}=5 x\] Rewriting this, we separate the variables as follows: \[3 y^{2} dy = 5 x dx\] This manipulation helps transform a differential equation into a form that can be straightforwardly integrated. This is the first step before moving on to integration. The goal is to make each side of the equation dependent on only one variable to facilitate easier calculation.

Integration is a fundamental concept in calculus used to find antiderivatives—functions whose derivatives yield the original function. Once you have applied separation of variables, each side of the differential equation can be integrated independently.

For the separated equation from our example, \[3 y^{2} dy = 5 x dx\] we proceed to integrate both sides.

• Left side (involving y): \[ \int 3 y^2 \, dy = \frac{3}{4} y^{4} \]
• Right side (involving x): \[ \int 5 x \, dx = \frac{5}{2} x^{2} \]

Combining both sides gives us: \[ \frac{3}{4} y^{4} = \frac{5}{2} x^{2} + C \] Here, C represents the constant of integration, reflecting that indefinite integrals can differ by a constant value.
Integrating both sides transforms the equation into one where the antiderivatives are equated, enabling us to solve for one of the variables.

Calculating solutions for differential equations involves a sequence of steps rooted in calculus principles like separation of variables and integration. Let's recap our problem and solution:

Our original equation: \[3 y^{2} \frac{d y}{d x} = 5 x\] Separating variables gave us: \[3 y^{2} dy = 5 x dx\] Integrating both sides produced: \[ \frac{3}{4} y^{4} = \frac{5}{2} x^{2} + C\] This last step indicates we have an implicit general solution combining x and y.

Practical calculus solutions often require integrating and using initial conditions to solve for constants like C. Without initial conditions in this problem, our final form is:
\[ \frac{y^{4}}{4} = \frac{5 x^{2}}{2} + C \]
Both separation of variables and integration are pivotal for solving ordinary differential equations, enabling you to manage and solve relations between changing quantities efficiently.