Verify that y is a solution of the differential equation. y=C e^k x, C any constant; (d y)/(d x)

step 1 We want to verify that Y is a solution of the given differential equation. Y equals C, E to the

Verify that y is a solution of the differential equation....

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Key Concepts

First-Order Linear Differential Equations

These are differential equations involving the first derivative of the unknown function and the function itself, often written in the form dy/dx + P(x)y = Q(x). They frequently model growth or decay processes and have well-established methods for finding general solutions.

Exponential Functions

Exponential functions, such as e^(kx), are fundamental in mathematics due to their unique property that the rate of change of the function is proportional to the function itself. This property makes them natural solutions to many differential equations, especially those involving growth or decay.

Chain Rule in Differentiation

The chain rule is a basic tool in calculus used to differentiate composite functions. In the context of exponential functions like e^(kx), it allows us to correctly compute the derivative by multiplying the derivative of the exponent (kx) by the derivative of the outer function (the exponential).

Verification of Solutions to Differential Equations

Verifying a solution involves substituting the proposed solution into the differential equation to check if the equation holds true. This process confirms that the function satisfies both the differential operator and any initial or boundary conditions present in the problem.

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