CadyMath

#1. \(\frac{dy}{dx} + y\frac{\arccos x}{\sqrt{1-x^2}} = 0,\quad |x|<1\)

\(y = Ce^{-\frac{3}{2}x^2}\)

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

\(y = Ce^{\frac{1}{2}(\arccos x)^2}\)

\(y = \frac{C}{x^2+4}\)

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

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

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

\(y = Ce^{-\frac{1}{2}(\arcsin x)^2}\)

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

#3. \(\frac{dy}{dx} + \frac{y}{x \ln x} = \frac{\sin(\ln x)}{x},\quad x > 1\)

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

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

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

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

#4. \(\frac{dy}{dx} + \frac{y}{x} = \frac{\ln(1+x)}{x^2},\quad x > 0\)

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

\(y = \frac{-1/\ln x + C}{\ln x}\)

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

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

#5. \(\frac{dy}{dx} - y\cot x = \sin x,\quad 0 < x < \pi\)

\( y=x(\ln x)^2+Cx \)

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

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

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

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

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

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

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

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