# Solving A Differential Equation

• Sane
In summary, the two given equations are y(1) = pi and dy/dx = (6x^2)/(2y + cos(y)). To solve for x, the integral of (2y + cos(y))dy is set equal to the integral of (6x^2)dx, resulting in y^2 + sin(y) = 2x^3. To exclude y, the constant of integration is needed and when x = 1, y = pi. The final solution is x = ((y^2 + sin(y) - pi^2)/2 + 1)^(1/3).
Sane
Given the two equations:

$$y(1) = \pi$$

$$\frac{dy}{dx} = \frac{6x^{2}}{2y + \cos{y}}$$

Solve for x:

\begin{align*} \int(2y+\cos{y})dy &= \int(6x^{2})dx\\ y^{2} + \sin{y} &= 2x^{3}\\ \end{align*}

But in order to set a value to the function y, I need some way to exclude y and then plug in 1 for x. How do I exclude y when it's in two separate terms?

Or is there an easier way to do this?

Last edited:
When x = 1, y = pi. Does that make it any simpler? Don't forget the constant of integration

Oh right! I forgot the C. That's why it wasn't working. I'm silly without my morning coffee... Haha. Thanks.

I just need to make sure I did this correctly.

\begin{align*} (\pi)^{2} + \sin{(\pi)} &= 2(1)^{3} + C\\ C &= \pi^{2} - 2\\ \\ 2x^{3} + \pi^{2} - 2 &= y^{2} + \sin{y}\\ 2x^{3} &= y^{2} + \sin{y} - \pi^{2} + 2\\ x &= (\frac{y^{2} + \sin{y} - \pi^{2}}{2} + 1)^{\frac{1}{3}} \end{align*}

Last edited:
I guess the step before the last would do just fine.

## What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model systems that change continuously over time or space.

## What is the process for solving a differential equation?

The process for solving a differential equation involves identifying the type of equation (such as linear, separable, or exact), finding a general solution, and then using initial conditions to determine a particular solution.

## What are the different methods for solving a differential equation?

Some common methods for solving differential equations include separation of variables, integrating factors, and substitution. Other techniques such as Laplace transforms, series solutions, and numerical methods may also be used depending on the complexity of the equation.

## What are the applications of solving differential equations?

Differential equations are used in a wide range of scientific fields, including physics, engineering, biology, and economics. They can be used to model real-world phenomena such as population growth, heat transfer, and fluid dynamics.

## What are the challenges of solving a differential equation?

Solving a differential equation can be challenging due to the complex mathematical concepts involved and the need for careful analysis and manipulation of equations. Some equations may also have multiple solutions or no analytical solutions at all, requiring the use of numerical methods.

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