Let u solve u′ + xu = 0 and suppose φ = uβ solves +2x+(x2+2)=0, with φ(0) = 1 and φ′(0) = 0. Find φ(x).

GATE 2020 MA – Question 43

Ordinary Differential Equations · Higher Order Equations and Systems · 2 marks · Multiple choice

Let u solve u′ + xu = 0 and suppose φ = uβ solves $\phi\prime\prime+2x\phi\prime+(x^2+2)\phi=0$, with φ(0) = 1 and φ′(0) = 0. Find φ(x).

$\cos^2x\,e^{-x^2/2}$
$\cos x\,e^{-x^2/2}$
$(1+x^2)e^{-x^2/2}$
$\cos x\,e^{-x^2}$

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Correct answer: (B) $\cos x\,e^{-x^2/2}$

Explanation

The first equation gives u proportional to $e^{-x^2/2}$. Substitute $\phi=e^{-x^2/2}\beta$ into the second equation; the β′ terms cancel and it reduces to β″ + β = 0. The initial conditions give β(0) = 1 and β′(0) = 0, hence β = cos x. Thus φ = cos x times $e^{-x^2/2}$, B.

GATE 2020 MA – Question 43

Explanation

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