91Ó°ÊÓ

A first-order ordinary differential equation (ODE) involves derivatives of the unknown function with respect to a single variable, typically denoted as x, and the highest derivative is of the first order. In the provided exercise, the equation given is \(\frac{d y}{d x}=4 x^{3} y\), involving the first derivative \( \frac{d y}{d x} \). First-order ODEs often appear in dynamic systems, modeling rates of change in various applications.
These equations are crucial because they form the foundation for more complex differential equations. Their solutions involve finding the function y(x) that satisfies the given equation. When solving such equations, one typically seeks general solutions, which involve a family of functions, or particular solutions that fit specific initial conditions or constraints.

Constant solutions in differential equations refer to solutions where the function y(x) remains constant for all values of x. In other words, the derivative of y with respect to x is zero. For the given differential equation \( \frac{d y}{d x}=4 x^{3} y \), we set the derivative to zero to find constant solutions. This leads to:
\ 4 x^{3} y = 0 \.
Since 4 and \( x^{3} \) cannot be zero for all x, the only solution is when y = 0. Therefore, y = 0 is a constant solution for this differential equation. Constant solutions are significant in understanding the behavior of differential systems as they can represent steady states or equilibrium points where the rate of change is zero.

Separable differential equations are equations where the variables can be separated on opposite sides of the equation. The given differential equation \( \frac{d y}{d x}=4 x^{3} y \) can be rewritten to separate the variables x and y:

\ \frac{1}{y} \frac{d y}{d x} = 4 x^{3} \.
By multiplying both sides by \( dx \) and integrating both sides, we obtain:

\ \frac{1}{y} d y = 4 x^{3} d x \.
Integrating, we get:

\ \ \int \frac{1}{y} d y = \ \int 4 x^{3} d x \.
The integral of \( \frac{1}{y} d y \) is \( \ \ln|y| \), and the integral of \( 4 x^{3} d x \) is \( x^{4} \). Adding the constant of integration, C, we get:

\( \ \ln|y| = x^{4} + C \).
Exponentiating both sides to solve for y gives:

\( y = e^{x^{4} + C} \).
Recognizing that \( e^C \) can be written as a new positive constant, K, we get:

\( y = Ke^{x^{4}} \). Integrating separable equations involves breaking the problem into more manageable parts, making it easier to find a solution. This method is one of the most straightforward techniques in solving first-order ODEs.