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Chapter 4: Problem 26

Solve the differential equation. $$ \frac{d y}{d x}=\frac{x-4}{\sqrt{x^{2}-8 x+1}} $$

Short Answer

Expert verified
The solution to the given differential equation is \(y = 2\sqrt{x^{2}-8x+1} + C\).

Step by step solution

01

Separating the variables

First, the differential equation needs to be separated into expressions containing x and y: \( dy = \frac{x-4}{\sqrt{x^{2}-8x+1}} dx \)
02

Integration

Now, both sides can be integrated: \(\displaystyle\int dy = \int \frac{x-4}{\sqrt{x^{2}-8x+1}} dx\). The left-hand side is straightforward, evaluating to y. But the right-hand side requires using a substitution method (with \(u = x^{2} - 8x + 1\)) to resolve the integral.
03

Completing the integral

After substituting \(u = x^{2} - 8x + 1\), the integral on the right-hand side simplifies to \(\displaystyle\int \frac{x-4}{\sqrt{u}} du\). This integral can be computed resulting in \(2\sqrt{u} + C\). You then replace the \(u\) value with \(x^2 - 8x + 1\) to get the final solution.
04

Final solution

Substitute \(u\) back into the equation to get the final solution: \(y = 2\sqrt{x^{2}-8x+1} + C\).

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Most popular questions from this chapter

In Exercises 35 and \(36,\) a model for a power cable suspended between two towers is given. (a) Graph the model, (b) find the heights of the cable at the towers and at the midpoint between the towers, and (c) find the slope of the model at the point where the cable meets the right-hand tower. \(y=10+15 \cosh \frac{x}{15}, \quad-15 \leq x \leq 15\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([a, b]\), then \(f\) is integrable on \([a, b]\).

Consider the function \(F(x)=\frac{1}{2} \int_{x}^{x+2} \frac{2}{t^{2}+1} d t\) (a) Write a short paragraph giving a geometric interpretation of the function \(F(x)\) relative to the function \(f(x)=\frac{2}{x^{2}+1}\) Use what you have written to guess the value of \(x\) that will make \(F\) maximum. (b) Perform the specified integration to find an alternative form of \(F(x)\). Use calculus to locate the value of \(x\) that will make \(F\) maximum and compare the result with your guess in part (a).

Find the integral. \(\int \frac{x}{x^{4}+1} d x\)

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