CadyMath

#1. \(\frac{dy}{dx} - y\frac{\sin x}{\cos x} = 0,\quad \cos x \neq 0\)

\(y = C\sec x\)

\(y = Ce^{-x(\ln x - 1)}\)

\(y = Ce^{e^{-x^2}}\)

\(y = \frac{C}{x^2}\)

#2. \(\frac{dy}{dx} - 2y = 0\)

\(y = Ce^{-\left(x + \frac{x^3}{3}\right)}\)

\(y = Ce^{2x}\)

\(y = Ce^{e^{-x^2}}\)

\(y = \frac{C}{x}\)

#3. \(x\frac{dy}{dx} - 3y = x^4e^x,\quad x > 0\)

\(y = \frac{\int \zeta(x) dx + C}{\ln x}\)

\(y = \frac{\text{PolyLog}(3,x) + C}{\ln x}\)

\(y = \frac{x^3/3 + x + C}{x + 1}\)

\(y = x^3(e^x + C)\)

#4. \(\frac{dy}{dx} + y \tan x = \sec^3 x,\quad -\pi/2 < x < \pi/2\)

\(y = \frac{-\frac{1}{2}e^{-x^2} + C}{x^2}\)

\(y = \frac{\frac{x}{2}\sqrt{x^2+1} + \frac{1}{2}\ln(x+\sqrt{x^2+1}) + C}{x}\)

\(y = \frac{\tan x + C}{\sec x}\)

\(y = \frac{x \ln(1+x^2) - 2x + 2\arctan x + C}{x}\)

#5. \(\frac{dy}{dx} - \frac{y}{x} = \frac{\ln x}{x},\quad x > 0\)

\(y = \frac{(\ln x)^2}{2} + Cx\)

\(y = \frac{x \text{Ci}(x) - \sin x + C}{x}\)

\(y = x(\sin x - x\cos x + C)\)

\(y=\frac{1}{2}-\frac{1}{x}+\frac{C}{x^2}\)

#6. \(\frac{dy}{dx} + \frac{y}{x} = \frac{\sin(\ln x)}{x},\quad x > 0\)

\(y = \frac{(\ln x)^2 + C}{2x}\)

\(y = \frac{-\cos(\ln x) + C}{x}\)

\(y = \frac{\frac{1}{2}\ln(1+x^2) + C}{1+x^2}\)

\(y = e^{-x}(\arctan x + C)\)