Answer to Question #233786 in Differential Equations for Phyroehan

Question

Find the general/particular solution of the following Differential Equations

(Exact D.E)

3.) [e^x + lny+(y/x)]dx [(x/y)+lnx+siny]dy=0

Answer 1

Solution;

Check for exactiness of the equation;

$M=e^x+lny+\frac yx$

$N=\frac xy+lnx+siny$

$\frac{dM}{dy}=\frac 1y+\frac 1x$

$\frac{dN}{dx}=\frac 1y+\frac 1x$

Clearly;

$\frac{dM}{dy}=\frac{dN}{dx}$

The equation is exact.

Solution is given by;

$\int_{y=c}Mdx+\int$ (terms in N independent of x)$dy=C$

$\int(e^x+lny+\frac yx)dx+\int sin (y )dy=C$

$\int e^xdx+lny\int1dx+y\int\frac 1xdx+\int sin (y)dy=C$

Solve to obtain the general solution;

$e^x+xln(y)+yln(x)-cos(y)=C$