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Chapter 1: Problem 47

\(d y / d x+2 y=0, y(0)=2, y(x)=A e^{-2 x}\)

Short Answer

Expert verified
The solution is \(y(x) = 2 e^{-2x}\).

Step by step solution

01

- Verify the Differential Equation

Substitute the given function, \(y(x) = A e^{-2x}\), into the differential equation \(\frac{dy}{dx} + 2y = 0\). Calculate the derivative \(\frac{dy}{dx}\) first. Since \(y(x) = A e^{-2x}\), we have \(\frac{dy}{dx} = -2A e^{-2x}\).
02

- Substitute and Simplify

Substitute \(y(x)\) and \(\frac{dy}{dx}\) into the differential equation: \(-2A e^{-2x} + 2A e^{-2x} = 0\). Simplify to verify that both sides equal zero: \(-2A e^{-2x} + 2A e^{-2x} = 0\). Thus, the given function satisfies the differential equation.
03

- Initial Condition

Use the initial condition to find the constant \(A\). The initial condition is \(y(0) = 2\). Substitute \(x = 0\) into the function: \(y(0) = A e^{0} = A\). We know from the initial condition that \(A = 2\).
04

- Final Solution

Substitute \(A = 2\) into the function to get the final solution: \(y(x) = 2 e^{-2x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Verification of Solution
To verify if a function is a solution to a differential equation, we need to substitute the function and its derivatives back into the original equation. For our exercise, we start with the given function,

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Most popular questions from this chapter

Write each of the following second order equations as a system of first order equations. (a) \(\frac{d^{2} x}{d t^{2}}-\frac{d x}{d t}-6 x=0\) (b) \(4 \frac{d^{2} x}{d t^{2}}+4 \frac{d x}{d t}+37 x=0\) (c) \(L \frac{d^{2} x}{d t^{2}}+g \sin x=0, L, g\) positive constants, \(x=x(t)\) (d) \(\frac{d^{2} x}{d t^{2}}-\mu\left(1-x^{2}\right) \frac{d x}{d t}+x=0, \mu>0\) con (e) \(t \frac{d^{2} x}{d t^{2}}+(b-t) \frac{d x}{d t}-a x=0, a, b\) constants

The function \(J_{1}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !(k+1) ! 2^{2 k+1}} \times\) \(x^{2 k+1}\) is called the Bessel function of order 1. Verify that \(J_{1}(x)\) is a solution of Bessel's equation of order \(1, x^{2} y^{\prime \prime}+x y+\) \(\left(x^{2}-1\right) y=0\).

\(d y / d x=\sqrt{x^{2}-16} / x\)

\(x\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+2 y=2 x\)

\(d y / d x=4 x^{3}-x+2, y(0)=1\)
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