Differential equation help

In summary: Remember that y'= u. So what is y?In summary, the given equation can be rewritten as a first order differential equation by letting u = dy/dx. Solving this equation and then using the fact that y' = u, we can find the solution for y.
  • #1
Froskoy
27
0

Homework Statement



Show that the equation [tex]-\frac{x}{2}\frac{dy}{dx} = \frac{d^2y}{dx^2}[/tex]

can be written as

[tex]\frac{d}{dx}\left({\ln \frac{dy}{dx}}\right) = -\frac{x}{2}[/tex]

3. Attempt at the solution

I approached this by writing [tex]\frac{d}{dx}\left({\frac{dy}{dx}}\right) = -\frac{x}{2}[/tex]

But this isn't the required result and I can't see how to get there?

Please help!

With very many thanks,

Froskoy.
 
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  • #2
As you know, if y = y(x), then d/dx ln(y) = y'/y. Thusly, dividing both sides by dy/dx gives you a very similar-looking form on the right side, which you should be able to solve from there.
 
  • #3
Froskoy said:

Homework Statement



Show that the equation [tex]-\frac{x}{2}\frac{dy}{dx} = \frac{d^2y}{dx^2}[/tex]

can be written as

[tex]\frac{d}{dx}\left({\ln \frac{dy}{dx}}\right) = -\frac{x}{2}[/tex]

3. Attempt at the solution

I approached this by writing [itex]\frac{d}{dx}\left({\frac{dy}{dx}}\right) = -\frac{x}{2}
How could you get from an equation that involves ln to one that does not but everything else is the same? You can't just erase the letters "ln"!

But this isn't the required result and I can't see how to get there?

Please help!

With very many thanks,

Froskoy.
If you let u= dy/dx, this becomes the first order equation
[tex]-\frac{x}{2}u= \frac{du}{dx}[/tex]
Can you solve that equation?

Once you know u, how do you find y?
 

What is a differential equation?

A differential equation is a mathematical equation that describes how a quantity changes over time or space. It involves the derivatives of the unknown function and can be used to model various real-world phenomena, such as population growth, motion of objects, and chemical reactions.

Why are differential equations important?

Differential equations are important because they provide a powerful tool for understanding and predicting the behavior of complex systems. They are widely used in physics, engineering, economics, and many other fields to create models and make predictions. Many natural phenomena can only be described using differential equations, making them an essential part of scientific research.

What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve a single independent variable, while PDEs involve multiple variables. SDEs involve random processes and are used to model systems that exhibit randomness or uncertainty.

How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some simple ODEs can be solved analytically using integration or other mathematical techniques. More complex equations may require numerical methods, such as Euler's method or Runge-Kutta methods. PDEs and SDEs often require advanced numerical techniques and computer simulations to find solutions.

What are some applications of differential equations?

Differential equations have countless applications in science, engineering, and economics. They are used to model physical systems such as fluid flow, electrical circuits, and climate change. They are also used in financial mathematics to model stock prices and other economic factors. Differential equations are also used in fields such as biology, chemistry, and neuroscience to study and understand complex biological systems.

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