# Do you dare

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)

Note by Akhilesh Prasad
2 years, 8 months ago

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## Comments

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Did 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.

- 2 years, 3 months ago

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@Rishabh Cool, can you please also solve this one if you have some time at your hand.

- 2 years, 4 months ago

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