Verify that the function f(t)=2 e^-t+t-1 is a solution of...

Verify that the function $f(t)=2 e^{-t}+t-1$ is a solution of the initial-value problem $y^{\prime}=t-y, y(0)=1 .$ [This is the function shown in Fig. $4(\mathrm{c}) .$ In Section $10.3,$ you will learn how to derive this solution.]

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Key Concepts

Ordinary Differential Equation (ODE)

An ordinary differential equation (ODE) is a relation involving a function and its derivatives. It describes how the value of the function changes with respect to one variable, typically time or space, and is fundamental in modeling physical phenomena, engineering problems, and other applications in mathematics.

Initial Value Problem (IVP)

An initial value problem is an ODE that comes with a specified value of the unknown function at a particular point, known as the initial condition. This condition is used to ensure the uniqueness of the solution and to model scenarios with a defined starting state.

Solution Verification

Solution verification involves substituting a proposed solution into the differential equation and the initial condition to check if they are satisfied. This process confirms that the function correctly represents the behavior dictated by the differential equation and meets the given initial parameters.

Differentiation of Functions

Differentiation is the process of computing the derivative of a function, which represents the rate of change of the function with respect to its variable. In verifying solutions for differential equations, accurate differentiation is crucial, as it ensures that the computed derivative matches the expression specified by the differential equation.

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