# Solving First Order Diff Eq: y^2 = x^2 + cx^3?

• Knissp
In summary, the conversation discusses a problem involving a differential equation and provides two potential solutions. The first solution is derived using substitution and integration, while the second solution is provided by a textbook. The individual seeking help confirms that their solution works and concludes that the textbook's answer is likely incorrect.
Knissp

## Homework Statement

dy/dx = (2x + y - 1)^2

## The Attempt at a Solution

Let u = 2x + y
du = 2 + dy/dx
dy/dx = du/dx - 2

dy/dx = (2x + y - 1)^2
so du/dx - 2 = (u-1)^2
du/dx = (u-1)^2 + 2
du / ((u-1)^2 + 2) = dx

1/sqrt(2) * arctan ((u-1)/sqrt(2)) = x + c

1/sqrt(2) * arctan ((2x + y -1)/sqrt(2)) = x + c

y = sqrt(2) * tan(sqrt(2) * x + C) + 1 - 2x

BUT the answer in the back of the textbook is y^2 = x^2 + cx^3. Did I mess up or is it a typo? Thank you.

You have two answers. Check them to see if they satisfy dy/dx = (2x + y - 1)^2. If you find that your answer satisfies this DE, that's pretty good evidence that the book's answer is wrong. I don't see anything obviously wrong with your work.

Cool thanks my solution worked.

If you feel really ambitious, you could check the book's solution. Sometimes with differential equations it's possible to get what look like completely different solutions, but they both work. The key is that they differ by a constant.

As an example, sin^2(x) and -cos^2(x) look to be very different, but differ only by a constant.

Yep the book's sol'n sure doesn't work. Thanks for the help! :)

## 1. What is a first order differential equation?

A first order differential equation is a mathematical equation that involves a function and its first derivative. It is used to model many physical and natural phenomena, such as growth and decay, motion, and chemical reactions.

## 2. How do you solve a first order differential equation?

To solve a first order differential equation, you need to find the function that satisfies the equation. This can be done by using analytical methods, such as separation of variables, integrating factors, or substitution, or by using numerical methods, such as Euler's method or the Runge-Kutta method.

## 3. What is a particular solution?

A particular solution is a specific solution to a differential equation that satisfies both the equation and any given initial conditions. It is unique for a given set of initial conditions.

## 4. What is a general solution?

A general solution is a family of solutions to a differential equation that includes all possible particular solutions. It contains one or more arbitrary constants, which can be determined by substituting in the initial conditions.

## 5. How do you solve for the arbitrary constants in a general solution?

The arbitrary constants in a general solution can be solved for by using the initial conditions. By substituting the values of the initial conditions into the general solution, you can solve for the constants and obtain a particular solution that satisfies the given initial conditions.

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