# Graphical represtation of a diff equation solution

• nhrock3
In summary, the theory states that the slope of a solution to a differential equation can be determined for any given point without knowing the actual solution. By drawing lines with the corresponding slopes, a photo can be created. The solution can be found by identifying a line that is tangent to all the lines in the photo. For the equation y'=y/x, the direction graph is shown in the provided photo. However, the process of creating this graph and finding the slopes at a specific point (x,y) is not explained. It is noted that x cannot equal 0 for the equation y/x. The collection of lines in the photo include y=x, y=2x, y=-x, etc. with corresponding slopes of 1,
nhrock3
the theory says that even without having the solution of the diff equation
for any given (x,y) we could know the slope of the solution at this point.

if we draw a line threw every point with the solpe of the solution we get the photo
if we find a line which is tangent to the lines then its the solution
if we found the solution then in each one of its points its tangent to the collection of the grath.

for this equation
y'=y/x
the direction graph is given by the following photo.
i don't know how they made this graph.
i don't have the idea of how to find the slopes at point (x,y)

we have y/x x cannot be 0 that's it

that line collection is some collection of lines y=x y=2x y=-x etc...
i don't know how it correspond to y/x

On the line y = x, y/x = 1, except when x = 0.
On the line y = 2x, y/x = 2, except when x = 0.
On the line y = kx, y/x = k, except when x = 0.
It's pretty straightforward.

thnks :)

## 1. What is a graphical representation of a differential equation solution?

A graphical representation of a differential equation solution is a visual representation of the relationship between the dependent and independent variables in a differential equation. It shows the behavior of the solution over a range of values for the independent variable.

## 2. How is a graphical representation of a differential equation solution created?

A graphical representation of a differential equation solution is created by plotting the values of the dependent variable against the values of the independent variable. This can be done by hand or using software such as MATLAB or Wolfram Alpha.

## 3. What information can be gained from a graphical representation of a differential equation solution?

A graphical representation of a differential equation solution can show the general behavior of the solution, such as whether it is increasing or decreasing, and if it has any critical points or asymptotes. It can also provide insight into the stability and long-term behavior of the solution.

## 4. How does the shape of the graph relate to the differential equation and its solution?

The shape of the graph is directly related to the equation and its solution. It can reveal information about the order and type of the differential equation, as well as the initial conditions and any parameters that may affect the behavior of the solution.

## 5. Can a graphical representation of a differential equation solution be used to solve the equation?

No, a graphical representation of a differential equation solution is not a method for solving the equation. It is a tool for visualizing and understanding the behavior of the solution, but other methods such as analytical or numerical techniques must be used to find the exact solution.

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