Solving differential equations of the form dy/dx = g(y) (HSC SSCE Mathematics Extension 1): Flashcards
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What do differential equations like dy/dx = g(y) represent?
What do differential equations like dy/dx = g(y) represent?
They show how a variable y changes based on itself.
What are steady state solutions in differential equations?
What are steady state solutions in differential equations?
Solutions where g(y_0) = 0, meaning y stays constant.
What is the purpose of separation of variables?
What is the purpose of separation of variables?
To isolate y and x for easier solving of equations.
How do you solve dy/dx = ky for y?
How do you solve dy/dx = ky for y?
Separate variables and integrate: y = Ce^(kt).
What does integration yield in solving differential equations?
What does integration yield in solving differential equations?
It provides ln|y| = kx + C after separation.
How do you find particular solutions in DEs?
How do you find particular solutions in DEs?
Use initial or boundary conditions to determine C.
What are the steady states for dy/dx = y^2/(1+y^3)?
What are the steady states for dy/dx = y^2/(1+y^3)?
Steady states are at y = 0 and y = -1 (not valid).
What is a common model for population growth?
What is a common model for population growth?
Logistic growth: dy/dx = ry(1-y/C).
How do you verify solutions of differential equations?
How do you verify solutions of differential equations?
Check if they satisfy the original equation and conditions.
Why are differential equations important?
Why are differential equations important?
They model growth, chemical reactions, and system behaviors.