Chapter 2: Q34E (page 77)

$x \equiv 0$ In problems 33-40, Solve the equation given in Problem 2.
${(y - 4 x - 1)}^{2} dx - dy = 0$

Short Answer

Expert verified

$\frac{1}{4} \log |\frac{y - 4 x - 3}{y - 4 x + 1}| - x = C$ and

Step by step solution

Solving the given equation

The given equation is,

${(y - 4 x - 1)}^{2} dx - dy = 0$

Simplify the above equation,

$\begin{array}{l} {(y - 4 x - 1)}^{2} dx = dy \\ \frac{dy}{dx} = {(y - 4 x - 1)}^{2} \cdot \cdot \cdot \cdot \cdot \cdot (1) \end{array}$

Let,

$\begin{array}{l} t = y - 4 x - 1 \\ \frac{dt}{dx} = \frac{dy}{dx} - 4 \\ \frac{dy}{dx} = \frac{dt}{dx} + 4 \end{array}$

Substitute the $t = y - 4 x - 1$ and $\frac{dy}{dx} = \frac{dt}{dx} + 4$ in the equation (1),

$\frac{dt}{dx} + 4 = {(t)}^{2}$

Simplify the above equation,

$\frac{dt}{dx} = t^{2} - 4$

Cross multiplication on both sides in the above equation,

$\frac{dt}{t^{2} - 4} = dx$

Integrating

Taking integration of both sides in the above equation

$\int \frac{dt}{t^{2} - 2^{2}} = \int dx$

One knows that,

$\int \frac{dx}{x^{2} - a^{2}} = \frac{1}{2 a} \log |\frac{x - a}{x + a}| + c$

Solve the above equation,

$\begin{array}{l} \frac{1}{2 (2)} \log |\frac{t - 2}{t + 2}| = x + c \\ \frac{1}{4} \log |\frac{t - 2}{t + 2}| - x = c \end{array}$

Finding the solution

Substitute the $t = y - 4 x - 1$ in the above equation,

$\frac{1}{4} \log |\frac{(y - 4 x - 1) - 2}{(y - 4 x - 1) + 2}| - x = c$

Simplify the above equation,

$\frac{1}{4} \log |\frac{y - 4 x - 3}{y - 4 x + 1}| - x = C$

Also, note that $x \equiv 0$ is a solution.

Hence the solution is $\frac{1}{4} \log |\frac{y - 4 x - 3}{y - 4 x + 1}| - x = C$ and $x \equiv 0$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$x \equiv 0$ In problems 33-40, Solve the equation given in Problem 2.
${(y - 4 x - 1)}^{2} dx - dy = 0$

Short Answer

Step by step solution

Solving the given equation

Integrating

Finding the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

x≡0In problems 33-40, Solve the equation given in Problem 2.y-4x-12dx-dy=0

Short Answer

Step by step solution

Solving the given equation

Integrating

Finding the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

$x \equiv 0$ In problems 33-40, Solve the equation given in Problem 2.
${(y - 4 x - 1)}^{2} dx - dy = 0$