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 $\begin{aligned} \log{\frac{dy}{dx}} &=8x+7y,\\ y(0)&=0? \end{aligned}$

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

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