Engineering Mathematics: Variable Separable Differential Equations

\begin{matrix} If the differential equation can be reduced to the general \\ form f (y) dy = ϕ (x) dx, then the solution is given by \\ int egrating both sides . \\ \int f (y) dy = \int ϕ (x) dx + c \\ Solved Problems \\ \begin{matrix} 1 . y \sqrt{1 - x^{2}} dy + x \sqrt{1 - y^{2}} dx = 0 & solution \\ 2 . (x^{2} - {yx}^{2}) \frac{dy}{dx} + y^{2} + {xy}^{2} = 0 & solution \\ 3 . \sec^{2} x \tan y dx + \sec^{2} y \tan x dy = 0 & solution \\ 4 . \frac{y}{x} \frac{dy}{dx} = \sqrt{1 + x^{2} + y^{2} + x^{2} y^{2}} & solution \\ 5 . e^{x} \tan y dx + (1 - e^{x}) \sec^{2} y dy = 0 & solution \\ 6 . \frac{dy}{dx} = {xe}^{y - x^{2}}, if y = 0 when x = 0 & solution \\ 7 . x \frac{dy}{dx} + \cot y = 0, if y = π / 4 when x = \sqrt{2} & solution \\ 8 . ({xy}^{2} + x) dx + ({yx}^{2} + y) dy = 0 & solution \\ 9 . \frac{dy}{dx} = e^{2 x - 3 y} + 4 x^{2} e^{- 3 y} & solution \\ 10 . y - x \frac{dy}{dx} = a (y^{2} + \frac{dy}{dx}) & solution \\ 11 . (x + 1) \frac{dy}{dx} + 1 = 2 e^{- y} & solution \\ 12 . (x - y)^{2} \frac{dy}{dx} = a^{2} & solution \\ 13 . (x + y + 1)^{2} \frac{dy}{dx} = 1 & solution \\ 14 . \sin^{- 1} (\frac{dy}{dx}) = x + y & solution \\ 15 . \frac{dy}{dx} = \cos (x + y + 1) & solution \\ 16 . \frac{dy}{dx} - x \tan (y - x) = 1 & solution \\ 17 . x^{4} \frac{dy}{dx} + x^{3} y + cosec (xy) = 0 & solution \end{matrix} \end{matrix}

Engineering Mathematics