Solve for the y-Intercept: Curve Through (4/9,1) with Slope of -2/y3

  • Thread starter Thread starter Slimsta
  • Start date Start date
  • Tags Tags
    Separable
Click For Summary
SUMMARY

The discussion focuses on finding the y-intercept of a curve defined by the differential equation dy/dx = -2/y³, passing through the point (4/9, 1). Participants clarify that the curve is not a straight line and must be solved using separable equations. The solution involves integrating to find the curve equation, resulting in y = sqrt4(-8x + 19). The final y-intercept, calculated by substituting x = 0, is approximately 2.087.

PREREQUISITES
  • Understanding of differential equations, specifically separable equations.
  • Knowledge of integration techniques for solving differential equations.
  • Familiarity with curve analysis and y-intercepts.
  • Basic algebra for manipulating equations and solving for constants.
NEXT STEPS
  • Study separable differential equations and their applications.
  • Practice integration techniques relevant to solving dy/dx equations.
  • Explore curve analysis, focusing on finding intercepts and behavior.
  • Review the concept of slope in relation to curves versus straight lines.
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations, and educators looking for examples of curve analysis and integration techniques.

Slimsta
Messages
189
Reaction score
0

Homework Statement


Find the y-intercept of the curve that passes through the point (4/9,1) and whose slope at (x,y) is −2/y3.


Homework Equations





The Attempt at a Solution


woudnt the y-int be y=-2x/y3 + 5.5 ?
i got it by plugging the slope into m in the equation y=mx+b
then plugged in the given points to get b..
but the question wants a numerical solution :S
 
Physics news on Phys.org
Slimsta said:
i got it by plugging the slope into m in the equation y=mx+b

That's the equation of a straight line, with constant slope, m. The slope you are given isn't constant, it depends on y. Your curve is not a straight line.

What is the slope of any curve f(x) at the point (x,f(x))?...Use that.
 
That makes no sense whatsoever. I think you know that. They want the y intercept of the curve. I think you have to solve for the curve. Whatever happened to the "separable equations" part of your post title? Shouldn't you separate dy/dx=(-2/y^3) and solve it??
 
Dick said:
That makes no sense whatsoever. I think you know that. They want the y intercept of the curve. I think you have to solve for the curve. Whatever happened to the "separable equations" part of your post title? Shouldn't you separate dy/dx=(-2/y^3) and solve it??

okay so i get dy/dx=(-2/y^3)
==> y^3 dy = -2dx
==> (y^4)/4 = -2x + C
==> y= sqrt4(-8x + 4C)

now what do i do?
 
Slimsta said:
okay so i get dy/dx=(-2/y^3)
==> y^3 dy = -2dx
==> (y^4)/4 = -2x + C
==> y= sqrt4(-8x + 4C)

now what do i do?

They gave you a point that it passes through, so find, the constant C.
 
rock.freak667 said:
They gave you a point that it passes through, so find, the constant C.

well if i plug (9/4, 1) in, i get C=4.75
from there y= sqrt4(-8x + 19)
now what do i do with this?
 
So you have a curve passing through the point given, with the slope that they require. What does the question tell you to do with it?
 
Office_Shredder said:
So you have a curve passing through the point given, with the slope that they require. What does the question tell you to do with it?

i got 2.087
once i got y = (-8x + 19)^0.25
y-int is when x=0 so y=19^0.25 = 2.087

its weird how the slope they gave us don't even help.. its there to make the question scary i guess.
 
You used the slope to figure out what y was as a function of x. How can you plug in x=0 if you don't know what the function is?
 

Similar threads

General Solution of inhomogeneous ODE
  • · Replies 2 ·
Replies
2
Views
2K
Calculus Review - Tangent of function and find y-intercept
  • · Replies 3 ·
Replies
3
Views
2K
Line passing through a point and normal to a curve
  • · Replies 4 ·
Replies
4
Views
2K
Calculating Area Under a Curve Using Change of Variables and Quotient Rule
  • · Replies 5 ·
Replies
5
Views
2K
Write the tangent lines to the curve
  • · Replies 1 ·
Replies
1
Views
1K
Find the equation of a tangent line to y = x^2?
  • · Replies 11 ·
Replies
11
Views
5K
Calculating the Tangent to a Parametric Curve at a Given Point
  • · Replies 3 ·
Replies
3
Views
2K
Simmons 7.10 & 7.11: Find Curves Intersecting at Angle pi/4
  • · Replies 1 ·
Replies
1
Views
2K
Finding x/y-intercepts, asymptotes, derivatives, and max min
  • · Replies 4 ·
Replies
4
Views
2K
Solving a separable matrix ODE.
  • · Replies 3 ·
Replies
3
Views
1K