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GATE 2023 CE (CE2) – Question 36

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 2 marks · Multiple choice

The solution of the differential equation $\frac{d^3y}{dx^3}-5.5\frac{d^2y}{dx^2}+9.5\frac{dy}{dx}-5y=0$ is expressed as $y=C_1e^{2.5x}+C_2e^{\alpha x}+C_3e^{\beta x}$, where $C_1,C_2,C_3,\alpha,\beta$ are constants, with α and β being distinct and not equal to 2.5. Which of the following options is correct for the values of α and β?

  1. 1 and 2
  2. −1 and −2
  3. 2 and 3
  4. −2 and −3

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Show answer and explanation

Correct answer: (A) 1 and 2

Explanation

**Step 1: characteristic equation.**
$$r^3-5.5r^2+9.5r-5=0 .$$

**Step 2: use the known root** $r=2.5$ and divide the polynomial by $(r-2.5)$:
$$r^3-5.5r^2+9.5r-5=(r-2.5)(r^2-3r+2).$$
(Check: $(r-2.5)(r^2-3r+2)=r^3-3r^2+2r-2.5r^2+7.5r-5=r^3-5.5r^2+9.5r-5$. ✓)

**Step 3: solve the quadratic.**
$$r^2-3r+2=(r-1)(r-2)=0\;\Rightarrow\;r=1,\ 2 .$$

So $\alpha,\beta=1$ and $2$ (option A).