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In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the con- stant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and high- light the one that satisfies the given initial condition.

17. y' + y = 0 y(x) = C * e ^ (- x) y(0) = 2

The plot above shows several solutions to the differential equation \(y' + y = 0\) with different values of \(C\). The red dashed line represents the specific solution \(y(x) = 2e^{-x}\) that satisfies the initial condition \(y(0) = 2\). Each curve represents a different exponential decay based on the constant \(C\).

### Would you like more details or have any questions?

Here are some related questions to explore:

1. What happens to the solution as \(x\) approaches infinity?
2. How would the solutions change if the differential equation were \(y' - y = 0\)?
3. What are the general characteristics of solutions to first-order linear differential equations?
4. How would you solve a first-order differential equation with a non-zero constant on the right-hand side, like \(y' + y = 1\)?
5. What methods can be used to solve first-order non-linear differential equations?

**Tip:** When solving differential equations, always verify your solution by substituting it back into the original equation to ensure it satisfies the equation.

Explore how to solve the differential equation y' + y = 0 with initial condition y(0) = 2. This problem involves understanding first-order differential equations and exponential functions. Learn the step-by-step solution and see graphical representations of different solutions based on the constant C.

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