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Question

# If parametric equations of the flow lines are $x = x(t), y = y(t)$ what differential equations do these functions satisfy? Deduce that $dy/dx = x$.

Solution

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At any point $\dfrac{dx}{dt}$ is the $x$ component of the velocity.

At any point $\dfrac{dy}{dt}$ is the $y$ component of the velocity.

Here it is given that Velocity at the point $(x, y$) is $\textbf{F}(x, y) = \textbf{ i }+x\textbf{ j }$

Note that, the $x$ component of the velocity is $1$ and

the $y$-component of the velocity is $x$

Therefore, we can write the following differential equations

$\dfrac{dx}{dt} = 1 \rightarrow \textbf{(1)}$

$\dfrac{dy}{dt} = x \rightarrow \textbf{(2)}$

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