CadyMath

#1. \(\frac{dy}{dx} + e^x y = 0\)

\(y = C(x^2-1)\)

\(y = C\csc^2 x\)

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

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

#2. \(\frac{dy}{dx} + y\frac{\cosh x}{\sinh x} = 0,\quad \sinh x \neq 0\)

\(y = C(1+x^3)\)

\(y = \frac{C}{1+\sin x}\)

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

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

#3. \( \sin x \frac{dy}{dx}+(\cos x)y=\tan x,\quad 0 < x < \pi/2 \)

\( y=(\csc x)(ln|\sec x|+C) \)

\( y=\frac{x^3}{3(x-1)^4}-\frac{x}{(x-1)^4}+\frac{C}{(x-1)^4} \)

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

\(y = \frac{x \text{Shi}(x) - \cosh x + C}{x}\)

#4. \(\frac{dy}{dx} + \frac{2x}{1+x^2}y = \frac{x}{(1+x^2)^2}\)

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

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

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

\(y = \sin x + C\csc x\)

#5. \(\frac{dy}{dx} + \frac{2y}{x} = \frac{\cos x}{x^2},\quad x > 0\)

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

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

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

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

#6. \(\frac{dy}{dx} + 2xy = 4x\)

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

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

\(y = \frac{\text{Ei}(x) - \frac{e^x}{x} + C}{x}\)

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