Let x=x(t) and y=y(t) be solutions of the differential equations (d x)/dt+a x=0 and (d y)/dt+by=0 respectively, a, b ∈ R. Given that x(0)=2 ; y(0)=1 and 3 y(1)=2 x(1), the value of t, for which x(t)=y(t), is :

Solution

Maths sir · 4 steps

Solve both as exponentials, then get eᵃ⁻ᵇ from the t = 1 condition as one block.

1. Solve each equation. dx/dt = -ax has solution x = x(0) e⁻ᵃᵗ. With x(0) = 2 and y(0) = 1, this gives x = 2e⁻ᵃᵗ and y = e⁻ᵇᵗ.
2. Use the condition. 3y(1) = 2x(1) becomes 3e⁻ᵇ = 4e⁻ᵃ. Multiply both sides by eᵃ and divide by 3 to get eᵃ⁻ᵇ = 4/3.
3. Set them equal. 2e⁻ᵃᵗ = e⁻ᵇᵗ gives 2 = e⁽ᵃ⁻ᵇ⁾ᵗ = (eᵃ⁻ᵇ)ᵗ. You never need a and b separately.

   (4/3)ᵗ = 2 ⇒ t = log_(4/3) 2

4. So. x(t) = y(t) when t = log_(4/3) 2, which is option (b).

The idea to keep. When two unknown constants appear only as a difference in the exponent, solve for eᵃ⁻ᵇ as one block. Do not try to find a and b separately.

Answer from the NTA final answer key. Solution by PrepWiser.