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Part CCSIR NET June 2024differentiate-the-radius-not-the-components

Differentiate the radius not the components

If x1=x1(t),x2=x2(t)x_{1} = x_{1}(t), x_{2} = x_{2}(t) is the solution of the initial value problem e^(−t) dx1/_{1}/dt =−x1+x2,e(−t)= -x_{1} + x_{2}, e^(-t) dx2/_{2}/dt =−x1−x2,x1(0)=1,x2(0)=0= -x_{1} - x_{2}, x_{1}(0) = 1, x_{2}(0) = 0, and r(t)=x12(t)+x22(t)r(t) = \sqrt{x_{1}^{2}(t) + x_{2}^{2}(t)}, then which of the following statements are true?

  1. A.r(t) → 0 as t→+∞t \to +\infty✓ Correct
  2. B.r(ln 2) = e⁻1^{1}✓ Correct
  3. C.r(ln 2) = 2e⁻1^{1}
  4. D.r(t)eᵗ → 0 as t→+∞t \to +\infty✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: W(f, g) ≡ 0 implies f and g are linearly dependent

More on this topic

The chapter behind this: Linear ODE, Wronskian and systems — free to read

From Ordinary Differential Equations › Linear ODE, Wronskian, variation of parameters, systems

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