AP Test Practice | DE Homework: Sketch Slope Field

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SUMMARY

The discussion centers on solving the differential equation \(\frac{dy}{dx} = 2x - y\) and sketching its slope field at specified points. Participants explore methods for separating variables and integrating the equation, questioning whether it qualifies as a homogeneous differential equation. The conversation highlights the need for clarity in transforming the equation into a solvable form, particularly through substitution or integrating factors. Ultimately, the focus is on effectively sketching the solution curve that passes through the point (0, 1).

PREREQUISITES
  • Understanding of differential equations, specifically first-order linear equations.
  • Familiarity with slope fields and their graphical representation.
  • Knowledge of integration techniques, including substitution and integrating factors.
  • Ability to manipulate and transform equations in terms of different variables.
NEXT STEPS
  • Study the method of integrating factors for solving first-order linear differential equations.
  • Learn about slope fields and how to sketch them for various differential equations.
  • Explore the concept of homogeneous differential equations and their characteristics.
  • Practice solving differential equations using substitution methods with various examples.
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations, as well as educators seeking to enhance their teaching methods in this area.

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Homework Statement



Consider the differential equation \frac{dy}{dx} = 2x - y

On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point (0, 1).

Homework Equations



\frac{du}{dx} = u where u = 2x - y

The Attempt at a Solution



\frac{dy}{dx} = 2x - y

\frac{du}{dx} = u where u = 2x - y

u' = 2 - \frac{dy}{dx}Then what do I do? I have no clue where to go from here.
Is this a homogeneous differential equation?

I know what to do in regards to the rest of the problem, I just don't know how to separate so I can integrate. Do I use the substitution method or integrating factors?
 
Physics news on Phys.org
du/dx isn't equal to u. dy/dx=u. What is du/dx? Try to write the differential equation completely in terms of u instead of y. Then separate it.
 

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