Ordinary differential equations

This will give you φ(x) and then you can take the derivative of that to show it satisfies the equation y'(x) = 1 + y(x)^2.In summary, we are trying to show that φ(x) is a solution of the differential equation y'(x) = 1 + y(x)^2. To do so, we must manipulate the given equation and eventually solve for φ(x). This can be done by multiplying both sides of the equation by (φ(x) + cot(x)) and then rearranging the terms to isolate φ(x). Once we have φ(x), we can take its derivative and show that it satisfies the given differential equation.
  • #1
tracedinair
50
0

Homework Statement



Show that φ(x) defined by,

(φ(x) - tan(x))/(φ(x) + cot(x)) = e^(∫(tan(x) + cot(x)) dx

is a solution of the differential equation y'(x) = 1 + y(x)^2

The Attempt at a Solution



Solving the right hand side first,

∫(tan(x) + cot(x) = ∫(tan(x)dx + ∫cot(x)dx = -ln|cos(x)| + ln|sin(x)|

e^(-ln|cos(x)| + ln|sin(x)|) = sin(x)/cos(x) = tan(x)

So,

(φ(x) - tan(x))/(φ(x) + cot(x)) = tan(x)

And here's where I get stuck. I cannot solve for phi. I just end up getting lost in the algebra.
 
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  • #2
Assuming your work so far is correct, aside from the fact that the absolute values disappeared in the ln terms, and the constant of integration is missing, you have this:

(φ(x) - tan(x))/(φ(x) + cot(x)) = tan(x)
Multiply both sides by (φ(x) + cot(x)):
φ(x) - cot(x) = tan(x) * (φ(x) + cot(x))

Multiply the right side, and then get both terms involving φ(x) on one side and all other terms on the other side. Factor φ(x) from the terms containing it, and divide both sides by the other factor.
 

Related to Ordinary differential equations

1. What is an ordinary differential equation (ODE)?

An ordinary differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves one independent variable and one or more dependent variables.

2. What are some real-world applications of ODEs?

ODEs can be used to model a wide range of physical phenomena, such as population growth, chemical reactions, and electrical circuits. They are also commonly used in engineering and economics to analyze and predict the behavior of systems.

3. How do you solve an ODE?

There are various methods for solving ODEs, including analytical methods such as separation of variables and substitution, and numerical methods such as Euler's method and Runge-Kutta methods. The appropriate method to use depends on the specific characteristics of the ODE.

4. What is the order of an ODE?

The order of an ODE is the highest order of derivative that appears in the equation. For example, a first-order ODE has only first derivatives, while a second-order ODE has second derivatives.

5. What is the difference between an ODE and a partial differential equation (PDE)?

The main difference between an ODE and a PDE is that an ODE involves only one independent variable, while a PDE involves more than one independent variable. This means that PDEs are typically used to model more complex systems and phenomena.

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