GATE 2020 EC Q.4: solution and answer

GATE 2020 · ECQ.4 · 1 mark · MCQ

The general solution of $\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 9 y = 0$ is

(A)
$y = C_{1} e^{3 x} + C_{2} e^{- 3 x}$
(B)
$y = (C_{1} + C_{2} x) e^{- 3 x}$
(C)
$y = (C_{1} + C_{2} x) e^{3 x}$
(D)
$y = C_{1} e^{3 x}$

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Answer and solution

ANSWER

The general solution is $y = (C_{1} + C_{2} x) e^{3 x}$ , option (C).

Official answer: CSource: Official answer key, GATE 2020 EC (IIT Delhi)

This is a linear homogeneous ODE with constant coefficients, so try $y = e^{m x}$ . Substituting gives the auxiliary (characteristic) equation $m^{2} - 6 m + 9 = 0$ .
Factorise: $m^{2} - 6 m + 9 = (m - 3)^{2} = 0$ , so $m = 3$ is a repeated real root (the discriminant is $36 - 36 = 0$ ).
For a repeated root $m$ , the two independent solutions are $e^{m x}$ and $x e^{m x}$ , so $y = (C_{1} + C_{2} x) e^{3 x} .$
Check the second solution. With $y = x e^{3 x}$ : $y^{'} = (1 + 3 x) e^{3 x}$ and $y^{''} = (6 + 9 x) e^{3 x}$ , so $y^{''} - 6 y^{'} + 9 y = [(6 + 9 x) - (6 + 18 x) + 9 x] e^{3 x} = 0.$
Rule out the other options. (A) uses the roots $m = \pm 3$ , which belong to $m^{2} - 9 = 0$ . (B) has the repeated-root form but with $m = - 3$ , which belongs to $(m + 3)^{2} = m^{2} + 6 m + 9 = 0$ . (D) is a solution, but it has only one arbitrary constant, and the general solution of a second-order ODE needs two. Answer: (C).

KEY IDEA

The auxiliary equation $m^{2} - 6 m + 9 = (m - 3)^{2} = 0$ has the repeated root $m = 3$ . A repeated root $m$ gives the pair $e^{m x}$ and $x e^{m x}$ , so the general solution is $(C_{1} + C_{2} x) e^{3 x}$ .

WATCH OUT FOR

Two mistakes are common. One is getting the sign of the root wrong: the middle term $- 6 d y / d x$ gives the root $m = + 3$ , not $- 3$ , so (B) is wrong. The other is writing $C_{1} e^{3 x} + C_{2} e^{3 x}$ for a repeated root. That is really a single constant times $e^{3 x}$ (the same as option D), so it cannot be the general solution of a second-order equation. The second solution must carry the factor $x$ .

From the GATE 2020 EC paper. Every question of this paper · Download the full paper · All 19 Differential equations questions, with notes

GATE 2020 EC Q.4: solution and answer

Answer and solution

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