Let u(x) and v(x) satisfy the differential equations

\(\frac { du }{ dx } +p(x)u=f(x)\) and \(\frac { dv }{ dx } +p(x)v=g(x)\), where \(p(x), f(x)\) and \(g(x)\) are continuous functions. If \(u({ x }_{ 1 })>v({ x }_{ 1 })\) for some \({ x }_{ 1 }\) and \(f(x)>g(x)\) for all \(x>{ x }_{ 1 }\), prove that any point \((x,y)\), where \(x>{ x }_{ 1 }\) does not satisfy the equations \(y=u(x)\) and \(y=v(x)\).

(This question has been asked in the IIT 97 Second Exam)

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TopNewestDid you get it eventually? All you have to show is that the two curves cannot intersect. From the linear differential equation, and the inequality, it should be straightforward. – Ameya Daigavane · 1 year, 4 months ago

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@Rishabh Cool, can you please also solve this one if you have some time at your hand. – Akhilesh Prasad · 1 year, 5 months ago

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