2.7   ODE No. 7

y(x)+y(x)cos(x)esin(x)=0

Mathematica : cpu = 0.0728113 (sec), leaf count = 23

DSolve[-E^(-Sin[x]) + Cos[x]*y[x] + Derivative[1][y][x] == 0,y[x],x]
 
{{y(x)xesin(x)+c1esin(x)}}

Maple : cpu = 0.006 (sec), leaf count = 13

dsolve(diff(y(x),x)+y(x)*cos(x)-exp(-sin(x)) = 0,y(x))
 
y(x)=(x+c1)esin(x)

Hand solution

(1)dydx+y(x)cos(x)=esin(x)

Integrating factor μ=ecosdx=esinx. Hence (1) becomes

ddx(μy(x))=μesin(x)

Replacing μ by esinx and integrating both sides

esinxy(x)=esinxesin(x)dx+Cesinxy(x)=dx+Cesinxy(x)=x+Cy(x)=xesinx+Cesin(x)