- #1
hotjohn
- 71
- 1
Homework Statement
dy/dx = (2x +y -1) / ( 4x -2y +1) , x= X +1 , y = Y-1 ,, how to make it into differential equation ? my ans is not same as the ans given .
P/s : in the second photo , it's lnx +c , sorry for the blur photo
in the photo posted , i have already showed that dy/ dY = 1 , dx/dX =1 , so i can conclude that dy=dY , dx=dX , so for the original dy/dx , i can make it as dY/dX , and replace the x as X+1 , and y = Y+1 , so i have dY/dX = (2X+Y) / (X+2Y)HallsofIvy said:What you give already is a differential equation! Do you mean "change from a differential equation in x and y to a differential equation in X and Y"? If you can immediately replace x and y on the right side of the equation by X+ 1 and Y- 1. For the left side, use the "chain" rule:
[tex]\frac{dy}{dx}= \frac{dy}{dY}\frac{dY}{dx}= \frac{dy}{dY}\frac{dY}{dX}\frac{dX}{dx}[/tex].
But , i still didnt get the anshotjohn said:in the photo posted , i have already showed that dy/ dY = 1 , dx/dX =1 , so i can conclude that dy=dY , dx=dX , so for the original dy/dx , i can make it as dY/dX , and replace the x as X+1 , and y = Y+1 , so i have dY/dX = (2X+Y) / (X+2Y)
The transformation of a function refers to the process of manipulating an existing function to create a new function. This is done by applying operations such as shifting, stretching, or reflecting the graph of the original function.
Functions are transformed in order to better understand their behavior and characteristics. Transformation can also help in solving equations and graphing complex functions.
There are four types of transformations for functions: translation, reflection, stretching, and shrinking. Translation involves shifting the graph horizontally or vertically. Reflection involves flipping the graph over an axis. Stretching and shrinking involve changing the scale of the graph.
Transformations can change the position, shape, and size of the graph of a function. They can also affect the domain and range of the function. For example, a horizontal translation will shift the graph left or right, while a vertical stretching will make the graph taller or shorter.
A translation is a specific type of transformation that involves shifting the graph of a function without changing its shape. Other transformations, such as reflections and stretches, will alter the shape of the graph in addition to moving it. Translations only change the position of the graph.