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 \[\begin{align} (8x^2-5yx^2)\frac{dy}{dx}+(5y^3+xy^3) &=0, \\ y(1) &=1? \end{align}\]
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 \[\log{\frac{dy}{dx}}=5x+2y, y(0)=0?\]