/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 \(\frac{d y}{d x}=\frac{-2(x+5)}... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 40

\(\frac{d y}{d x}=\frac{-2(x+5)}{(x+2)(x-4)}\)

Short Answer

Expert verified
y = \text{ln}\bigg| \frac{(x-4)}{(x+2)^3} \bigg| + C

Step by step solution

01

Separate the Variables

Isolate the variables on opposite sides of the equation. The equation is already in a form suitable for integration:\[ \frac{d y}{d x} = \frac{-2(x+5)}{(x+2)(x-4)} \]
02

Integrate Both Sides

Integrate both sides with respect to their respective variables. We need to find the antiderivative of both sides:\[ \frac{dy}{dx} = \frac{-2(x+5)}{(x+2)(x-4)} \]
03

Simplify the Integral

To solve this integral, use partial fraction decomposition. Express the fraction as:\[ \frac{-2(x+5)}{(x+2)(x-4)} = \frac{A}{x+2} + \frac{B}{x-4} \]Solving for A and B results in:\[ A = -3 \]\[ B = 1 \]So the integral becomes:\[ \frac{dy}{dx} = -\frac{3}{x+2} + \frac{1}{x-4} \]
04

Integrate Each Term

Integrate each term individually:\[ \frac{dy}{dx} = -3 \frac{1}{x+2} + 1 \frac{1}{x-4} \]The antiderivative is:\[ y = -3 \text{ln}|x+2| + \text{ln}|x-4| + C \]
05

Simplify the Solution

Combine the logarithmic terms and simplify the expression:\[ y = \text{ln}\bigg| \frac{(x-4)}{(x+2)^3} \bigg| + C \]

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Key Concepts

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

separation of variables
Separation of variables is a technique used to solve certain differential equations. The goal is to rearrange the equation so that each variable and its differential are on opposite sides of the equation. This makes it easier to integrate each side separately. For instance, given the differential equation \( \frac{d y}{d x} = \frac{-2(x+5)}{(x+2)(x-4)} \), we can separate the variables by isolating \( dy \) and \( dx \) on different sides of the equation. This structure sets us up perfectly for the next step: integration.
integration
Integration is the process of finding the antiderivative or integral of a function. It helps us determine the accumulation of quantities, which is particularly useful in finding areas under curves and solving differential equations. After separating the variables in the equation \( \frac{dy}{dx} = \frac{-2(x+5)}{(x+2)(x-4)} \), we need to integrate both sides. The integration of \( dy \) is straightforward, resulting in \( y + C \) where \( C \) is the integration constant. However, integrating the right side requires more effort and specific techniques.
partial fraction decomposition
Partial fraction decomposition is a method to

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Most popular questions from this chapter

Show that \(u(x, y)=\tan ^{-1}(y / x)\) satisfies Laplace's equation \(u_{x x}+u_{y y}=0\).

\(d y / d x-y=\sin x, y=\left(e^{x}-\cos x-\sin x\right) / 2\)

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

\(3 y\left(t^{2}+y\right) d t+t\left(t^{2}+6 y\right) d y=0, t^{3} y+3 t y^{2}=\) \(8, t=2\)

\(x^{2} y^{\prime \prime}+3 x y^{\prime}+2 y=0, y=x^{-1}(\cos (\ln x)-\) \(\sin (\ln x))\)
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