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These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

What is the solution of the differential equation

\[4\left(\frac{dx}{dy}\right)=8x-y^2?\]

**Details and assumptions**

Use \(C\) as the constant of integration.

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What is the solution of the differential equation \[ \frac{1}{5}\left(\frac{1}{{x}^{3}}\frac{dx}{dy}-\frac{1}{{x}^{2}}y\right) =y\sin{{y}^{2}} ?\]

**Details and assumptions**

Use \(C\) as the constant of integration.

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What is the solution of the differential equation \[\begin{align} \log{\frac{dy}{dx}} &=8x+7y,\\ y(0)&=0? \end{align}\]

**Note:** In this problem, \(\log(x)\) is the natural logarithm (log base e) of \(x.\)

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What is the solution of the differential equation \[\log{\frac{dy}{dx}}=5x+2y, y(0)=0?\]

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