Which differential equation, (I)-(VI), has the function as a...

Which differential equation, (I)-(VI), has the function as a solution?
I. $\quad y^{\prime}=-2 x y$
II. $\quad y^{\prime}=-x y$
III. $\quad y^{\prime}=x y$
IV. $\quad y^{\prime}=-x^{2} y$
V. $\quad y^{\prime}=x^{-3} y$
VI. $\quad y^{\prime}=2 x y$
$$y=e^{-x^{2}}$$

Key Concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) involve functions of a single variable and their derivatives. They are used to model problems where the rate of change of a quantity depends on the current state of that quantity and possibly on the independent variable, and finding functions that satisfy these relationships is central to many applications in science and engineering.

Verification of Solutions

Verifying a solution to a differential equation involves substituting the proposed function into the equation and confirming that both sides of the equation are equal for all values within the domain. This process ensures that the function truly satisfies the differential equation and is a valid solution.

Chain Rule in Differentiation

The chain rule is a fundamental tool in calculus used to differentiate composite functions. When a function is composed of an outer function and an inner function, the chain rule states that the derivative is the product of the derivative of the outer function (evaluated at the inner function) and the derivative of the inner function. This rule is crucial when differentiating functions where the variable is within another function, such as exponential functions with variable exponents.

Exponential Functions

Exponential functions, which typically have the form exp(g(x)), are important in differential equations because their derivatives often maintain a similar functional form to the original function. This property leads to them being natural candidates for solutions of linear differential equations, where the rate of change is proportional to the function itself.

Transcript

Please give Ace some feedback