My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

LinkBack Thread Tools Display Modes
June 20th, 2016, 09:24 AM   #1
Joined: Jun 2016
From: sri lanka

Posts: 3
Thanks: 0

geometrical problem to solve using differential equation..

How to get the equation for this using differential equations... Find the curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact....

Last edited by skipjack; June 20th, 2016 at 09:29 AM.
madhawavish is offline  
June 21st, 2016, 04:28 AM   #2
Math Team
Joined: Jan 2015
From: Alabama

Posts: 3,264
Thanks: 902

Have you not even started? Call the function y(x). The tangent line to the graph y= y(x) at point $\displaystyle (x_0, y_0)$ is $\displaystyle y= y'(x_0)(x- x_0)+ y_0$.
When x= 0, $\displaystyle y= y'(x_0)(0- x_0)+ y_0= y_0- x_0y'(x_0)$ so the tangent line will intersect the y-axis at $\displaystyle (0, y_0- x_0y'(x_0))$. When y= 0, $\displaystyle 0= y'(x_0)(x- x_0)+ y_0$, $\displaystyle y'(x_0)(x- x_0)= -y_0$, $\displaystyle x- x_0= -\frac{y_0}{y'(x_0)}$, and $\displaystyle x= x_0- \frac{y_0}{y'(x_0)}$. So the tangent line intersects the x-axis at $\displaystyle \left(x_0- \frac{y_0}{y'(x_0)}, 0\right)$.

The point mid-way between the two intersections is $\displaystyle \left(\frac{x_0}{2}- \frac{y_0}{2y'(x_0)}, \frac{y_0}{2}- \frac{x_0y'(x_0)}{2}\right)$. We are told that the point of tangency, $\displaystyle (x_0, y_0)$ is that midpoint so we must have $\displaystyle x_0= \frac{x_0}{2}- \frac{y_0}{2y'(x_0)}$ or $\displaystyle x_0= -\frac{y_0}{y'(x_0)}$ and $\displaystyle y_0= \frac{y_0}{2}- \frac{y'(x_0)}{2}$ or $\displaystyle y_0= y'(x_0)$.

Since that is to be true at any point of tangency, we must have $\displaystyle x= -\frac{y}{y'(x)}$ or $\displaystyle xy'(x)= -y$ and $\displaystyle y= y'(x)$.
Country Boy is offline  

  My Math Forum > College Math Forum > Calculus

differential, equation, geomatrical, geometrical, problem, solve

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Can anybody help solve this differential equation? DanFa1996 Calculus 1 December 7th, 2015 11:32 AM
Please help me solve this geometrical & exponential distribution problem Pcoppus Advanced Statistics 1 September 13th, 2015 10:57 AM
solve the differential equation ?? esc30 Differential Equations 2 November 2nd, 2013 04:55 AM
Solve this differential equation. gen_shao Differential Equations 4 September 23rd, 2013 04:52 AM
Solve the differential equation sivela Differential Equations 3 February 5th, 2012 09:02 PM

Copyright © 2019 My Math Forum. All rights reserved.