(a) Write down the exact solution to the initial value...

(a) Write down the exact solution to the initial value problem $\frac{d u}{d t}=\frac{2}{7} u, u(0)=\frac{1}{3}$. (b) Suppose you make the approximation $u(0)=.3333$. At what point does your solution differ from the true solution by 1 unit? by 1000 units? (c) Answer the same question if you also approximate the coefficient in the differential equation by $\frac{d u}{d t}=.2857 u$.

(a) Write down the exact solution to the initial value problem $\frac{d u}{d t}=\frac{2}{7} u, u(0)=\frac{1}{3}$.
(b) Suppose you make the approximation $u(0)=.3333$. At what point does your solution differ from the true solution by 1 unit? by 1000 units? (c) Answer the same question if you also approximate the coefficient in the differential equation by $\frac{d u}{d t}=.2857 u$.

Key Concepts

Exponential Growth Differential Equations

Differential equations of the form u'(t) = ku represent models of exponential growth or decay, where k is a constant. The solution is an exponential function that quickly demonstrates either rapid increase or decrease depending on the sign and magnitude of k. This form of equation is widely used in many fields such as physics, biology, and finance to describe systems that change proportionally to their current state.

Initial Value Problems

An initial value problem in differential equations involves finding a solution from a given differential equation that also satisfies a specified initial condition at a particular time point. This initial condition uniquely determines the constant of integration in the solution, ensuring that the solution fits the precise scenario described by the initial state.

Exact Solutions of Ordinary Differential Equations

Exact solutions refer to expressions obtained by directly integrating the differential equation, often using separation of variables, which yield a precise formula for the dependent variable in terms of the independent variable. For first-order linear differential equations, the integrating factor method or simple separation is used, resulting in a solution that can be expressed in terms of elementary functions.

Approximation and Error Analysis in Differential Equations

Approximation and error analysis techniques assess how small changes in parameters, such as initial conditions or coefficients, can lead to significant differences in the solutions of differential equations. This is particularly important for sensitive models like exponential growth, where even minor approximations can result in substantial deviations over time. Understanding this sensitivity helps in evaluating the reliability of numerical approximations and in making decisions on the level of precision required.

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