Chapter 2: Q20E (page 70)

Verify that when the linear differential equation $[P (x) y - Q (x)] dx + dy = 0$ is multiplied by $μ (x) = e^{\int P (x) dx}$ , the result is exact.

Short Answer

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Step by step solution

General form of special integrating factors

Theorem 3:

If $\frac{(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})}{N}$ is continuous and depends only on x, then $μ (x) = \exp [\int (\frac{(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})}{N}) dx]$

is an integrating factor for the equation.

If $\frac{(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y})}{M}$ is continuous and depends only on y, then $μ (y) = \exp [\int (\frac{(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y})}{M}) dy]$

is an integrating factor for the equation.

Evaluation method

Given that, $[P (x) y - Q (x)] dx + dy = 0 \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Multiply by $μ (x) = e^{\int P (x) dx}$ on both sides. We get,

$[P (x) {ye}^{\int P (x) dx} - Q (x) e^{\int P (x) dx}] dx + e^{\int P (x) dx} dy = 0 \cdot \cdot \cdot \cdot \cdot \cdot (2)$ $[P (x) {ye}^{\int P (x) dx} - Q (x) e^{\int P (x) dx}] dx + e^{\int P (x) dx} dy = 0 \cdot \cdot \cdot \cdot \cdot \cdot (2)$

Let $M = P (x) {ye}^{\int P (x) dx} - Q (x) e^{\int P (x) dx}, N = e^{\int P (x) dx}$ .

Then,

$\begin{array}{l} \frac{\partial M}{\partial y} = P (x) e^{\int P (x) dx} \\ \frac{\partial N}{\partial x} = P (x) e^{\int P (x) dx} \end{array}$

Therefore, .

$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

So, the result for this problem is exact.

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Verify that when the linear differential equation $[P (x) y - Q (x)] dx + dy = 0$ is multiplied by $μ (x) = e^{\int P (x) dx}$ , the result is exact.

Short Answer

Step by step solution

General form of special integrating factors

Evaluation method

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Verify that when the linear differential equation Pxy-Qxdx+dy=0is multiplied by μx=e∫Pxdx, the result is exact.

Short Answer

Step by step solution

General form of special integrating factors

Evaluation method

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Verify that when the linear differential equation $[P (x) y - Q (x)] dx + dy = 0$ is multiplied by $μ (x) = e^{\int P (x) dx}$ , the result is exact.