u t=u xx on 0 0, with u(0,t)=u(π,t)=0 and u(x,0)=sin4x cos3x. Find u(π/4,t).

$e^{-49t}(1-e^{48t})/(2\sqrt2)$

$e^{-49t}(1+e^{48t})/(2\sqrt2)$

$e^{-49t}(1-e^{48t})/(4\sqrt2)$

GATE 2025 MA – Question 20

Partial Differential Equations · Heat and Wave Equations · 1 mark · Multiple choice

u_t=u_xx on 0<x<π,t>0, with u(0,t)=u(π,t)=0 and u(x,0)=sin4x cos3x. Find u(π/4,t).

$e^{-49t}(e^{48t}-1)/(2\sqrt2)$
$e^{-49t}(1-e^{48t})/(2\sqrt2)$
$e^{-49t}(1+e^{48t})/(2\sqrt2)$
$e^{-49t}(1-e^{48t})/(4\sqrt2)$

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Correct answer: (A) $e^{-49t}(e^{48t}-1)/(2\sqrt2)$

Explanation

Product-to-sum gives initial data (sin7x+sinx)/2. Hence $u=(e^{-49t}\sin7x+e^{-t}\sin x)/2$. At x=π/4, sine values are −1/√2 and +1/√2, giving $(e^{-t}-e^{-49t})/(2\sqrt2)$, **A**.

GATE 2025 MA – Question 20

Explanation

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