Consider the following initial$-$value problem.

$y\frac{dx}{dy}-x=4y^2,x(7)=1$

Find the coefficient function $P(y)$ when the given differential equation is written in the standard form $\frac{dx}{dy}+P(y)x=f(y).$

$P(y)=$

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Find the integrating factor for the differential equation.

$e^{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}P(y)dy}=$

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Solve the given initial$-$value problem.

$x(y)=$

Give the largest interval I over which the solution is defined for the given initial condition. $($Enter your answer using interval notation.$)$

$I=$

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Solve the given initial$-$value problem.