Exercise · ZW-EQUILIBRIUM-POINTS-009 · ★★★ Advanced
If \(y^*\) is an equilibrium and \(y(0)=y^*\), what is the solution of the initial value problem?
Choose one option.
Show answer
- B. Constant: \(y(t)=y^*\) for all \(t\)
Explanation
The constant function \(y^*\) solves the equation and the initial value; by uniqueness it is the only solution.
Review the concept: Equilibrium points
An equilibrium point of an autonomous equation \(y'=f(y)\) is a constant value \(y^*\) of the unknown at which \(f\) vanishes.
\[f(y^*)=0\;\Longleftrightarrow\; y(t)\equiv y^*\text{ is a solution}\]
More practice problems: equilibrium points
- ZW-EQUILIBRIUM-POINTS-001 Find the equilibrium point of y'=3y-6.
- ZW-EQUILIBRIUM-POINTS-002 What are the equilibria of y'=y^2-4y+3?
- ZW-EQUILIBRIUM-POINTS-003 How many equilibria does y'=y(y-2)(y+1) have?
- ZW-EQUILIBRIUM-POINTS-004 Which of these equations is autonomous?
- ZW-EQUILIBRIUM-POINTS-005 How many real equilibria does y'=y^2+1 have?
- ZW-EQUILIBRIUM-POINTS-006 How many equilibria does y'=sin y have in the interval [0,2π]?
- ZW-EQUILIBRIUM-POINTS-007 Find the equilibria of y'=2y(1-tfracy10).
- ZW-EQUILIBRIUM-POINTS-008 A tank satisfies y'=12-4y. At what value does it stay constant?