Exercise · ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-010 · ★★★ Advanced
How many equilibrium points does \(x'=x(1-y),\ y'=y(x-1)\) have?
Choose one option.
Show answer
- D. \(2\)
Explanation
If \(x(1-y)=0\) and \(y(x-1)=0\): \(x=0\) forces \(y=0\), and \(y=1\) forces \(x=1\). The equilibria are \((0,0)\) and \((1,1)\).
Review the concept: System of differential equations
A set of \(n\) equations that fix, through a vector function \(\mathbf F\), the derivatives of \(n\) unknown functions \(x_1(t),\dots,x_n(t)\) of a single variable \(t\).
\[\mathbf x'=\mathbf F(t,\mathbf x),\qquad \mathbf x=(x_1,\dots,x_n)\]
More practice problems: system of differential equations
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-001 With x_1=y and x_2=y', the equation y''-3y'+2y=0 is equivalent to mathbf x'=Amathbf x. What is A?
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-002 How many equations does the first-order system equivalent to a third-order equation have?
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-003 The system x'=y,\ y'=4x has the solution:
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-004 Find the equilibrium point of x'=x-y,\ y'=x+y-2.
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-005 Solve x'=2x,\ y'=-3y with x(0)=1,\ y(0)=2.
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-006 Turn y''+4y=0 into a system with x_1=y,\ x_2=y'. Which system is it?
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-007 If x'=y and y'=-x, what is x''?
- ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-008 How many initial conditions determine a unique solution of a system of 3 first-order equations?