Exercise · ZW-SYSTEM-OF-DIFFERENTIAL-EQUATIONS-009 · ★★★ Advanced
Which of these systems is decoupled (each equation involves a single unknown)?
Choose one option.
Show answer
- B. \(x'=2x,\ y'=3y\)
Explanation
Only in \(x'=2x,\ y'=3y\) does \(x'\) depend on \(x\) alone and \(y'\) on \(y\) alone. The others have cross terms, which couple the unknowns.
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
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- 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?
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