Calculus
# First Order Differential Equations

What is the solution of the following differential equation: \[\left(\frac{dy}{dx}\right)^{5}-x\frac{dy}{dx}+y=0?\]

**Note:** \(C\) is a constant.

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.

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 \(x.\)

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

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