Question

Verifying a Particular Solution In Exercises \(9-12\) , verify the particular solution of the differential equation. $$ \begin{array}{rl}{y=4 e^{-6 x^{2}}} & {y^{\prime}=-12 x y} \\ {} & {y(0)=4}\end{array} $$

Short answer

After substitution and simplification, the given function \(y=4 e^{-6 x^{2}}\) and its derivative \(y^{\prime}=-12 x y\) satisfy the differential equation, hence it's confirmed that this \(y\) is indeed a solution to the given differential equation.

Step-by-step solution

Identify the given function and its derivative

The given function is \(y=4 e^{-6 x^{2}}\) and its derivative is \(y^{\prime}=-12 x y\).

Substitute the function into the differential equation

Substitute \(y\) and \(y^{\prime}\) into the equation \( y^{\prime} = -12 x y \) we get: \[ -12x(4 e^{-6 x^{2}})=-12 x y \]

Simplify

After simplifying, we get \[ -48x e^{-6 x^{2}} = -48x e^{-6 x^{2}} \]

Verify

From step 3, we see that our simplified expression conforms to our given \( y^{\prime} = -12 x y \). Hence this concludes that the proposed function \( y \) is indeed a solution to the given differential equation.

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. In simple terms, it represents how a particular quantity changes over time or another variable. These equations often model real-world phenomena such as motion, growth, decay, and various engineering systems.

To solve a differential equation, one must find the function that satisfies the equation for all values of the variables involved. This process usually involves integration, separation of variables, substitution, or other methods specific to the type of differential equation. Differential equations can be classified into ordinary differential equations (ODEs), where there is only one independent variable, and partial differential equations (PDEs), where the function depends on multiple independent variables.

Particular Solutions

While a general solution to a differential equation represents an entire family of functions, a particular solution is a specific member of that family that satisfies the differential equation as well as an initial condition or boundary condition. In the context of your textbook exercise, the given function is a particular solution because it is stated to satisfy both the differential equation and the initial condition, \( y(0)=4 \).

Identifying a particular solution involves applying the initial conditions to the general solution to solve for any constants that may be part of the general solution. These types of solutions are critical when solving practical problems where conditions are specified at a particular point or interval.

Derivatives

Derivatives represent the rate of change of a function with respect to its variable. They are a foundational tool in calculus, providing essential information about the behavior of the function, such as increasing or decreasing trends, maximum and minimum points, and the shape of its graph. For a function \( y=f(x) \), the derivative of \( y \) with respect to \( x \) is often denoted by \( y' \) or \( \frac{dy}{dx} \).

In your exercise, the differential equation involves the derivative \( y' = -12xy \), which suggests that the rate at which \( y \) changes is directly proportional to \( -12x \) times the function \( y \). This relationship is part of what you are asked to verify using the solution provided.

Exponential Functions

Exponential functions have the form \( y=a e^{kt} \) where \( a \) and \( k \) are constants, \( e \) is the base of the natural logarithm, and the variable \( t \) often represents time. These functions are used in many applications because they describe growth or decay processes, like population growth or radioactive decay.

The function given in your exercise, \( y=4 e^{-6 x^{2}} \), is an exponential function where the power of \( e \) includes a squared term, suggesting the rate of decay is not constant but changes with \( x \). This is a less common form of an exponential function in simple models, showing the depth and variety of behaviors that exponential functions can represent.