Let \( u(x) \) and \( v(x) \) satisfy the differential equations \( \frac{d u}{d x}+p(x) u=f(x) ...

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Let \( u(x) \) and \( v(x) \) satisfy the differential equations \( \frac{d u}{d x}+p(x) u=f(x) \) and \( \frac{d v}{d x}+p(x) v=g(x) \), where \( p(x), f(x) \) and \( g(x) \) are continuous functions. If \( u\left(x_{1}\right)v\left(x_{1}\right) \) for some \( x_{1} \) and \( f(x)g(x) \) for all \( xx_{1} \), prove that any point \( (x, y) \) where \( xx_{1} \) does not satisfy the equations \( y=u(x) \) and \( y= \) \( v(x) \)
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