Question

Solve the following ordinary differential equation:
{y(1+(1/x))+cosy}dx +(x+logx-xsiny)dy=0

Answer 1

Answer on Question #86242 – Math – Differential Equations

Question

Solve the following ordinary differential equation:

$$\left\{y \left(1 + \frac {1}{x}\right) + \cos y \right\} d x + (x + \log x - x \sin y) d y = 0$$

Solution

$$M (x, y) = y \left(1 + \frac {1}{x}\right) + \cos y, \frac {\partial M}{\partial y} = 1 + \frac {1}{x} - \sin y$$ $$N (x, y) = x + \log x - x \sin y, \frac {\partial N}{\partial x} = 1 + \frac {1}{x} - \sin y$$ $$\frac {\partial M}{\partial y} = \frac {\partial N}{\partial x}$$

Exact Differential Equation

$$f (x, y) = \int M (x, y) d x = \int \left(y \left(1 + \frac {1}{x}\right) + \cos y\right) d x =$$ $$= y (x + \log x) + x \cos y + g (y)$$ $$\frac {\partial f}{\partial y} = x + \log x - x \sin y + \frac {d}{d y} (g (y)) = N (x, y)$$ $$x + \log x - x \sin y + \frac {d}{d y} (g (y)) = x + \log x - x \sin y$$ $$\frac {d}{d y} (g (y)) = 0$$ $$g (y) = c _ {1}$$ $$f (x, y) = y (x + \log x) + x \cos y + c _ {1} = c _ {2}$$

Finally, the solution of the ordinary differential equation is given by

$$f (x, y) = C,$$

where $f(x,y) = y(x + \log x) + x\cos y$, $C$ is an arbitrary real constant.

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