# Solve differential equation sin(x) dy/dx+(cos(x))y=0, y((7pi)/6)=-2

Question
Solve differential equation $$sin(x) dy/dx+(cos(x))y=0$$, $$y((7pi)/6)=-2$$

2020-11-15
$$sinx dy/dx+ ycosx=0$$
$$sinx dy= -ycosx dx$$
$$dy/y= - cosx/sinx dx$$
$$int dy/y= -int cosx/sinx dx$$
$$ln abs(y)= -ln abs(sinx)+ln abs(C)$$
$$lny= ln (C/sinx)$$
$$ysinx= C$$
Now, We are applying the given Initial Condition is as follow
$$y((7pi)/6)= -2$$
$$-2*sin((7pi)/6)= C$$ $$-2* -1= C$$ $$:.sin(7pi/6)= -1$$
C=2
$$ysinx= 2$$

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