Solve the differential equation. Please show ALL working clearly and legibly and

Question

Solve the differential equation. Please show ALL working clearly and legibly and explain steps as this is a difficult problem to understand. Section is on exact equations. Thanks much for all your help!

(1 point) Solve the following differential equation: 1point) Solve the fo (tan(x)-5 sin(x) sin(y))dz + 5 cos(z) cos(y)dy-0 cOS(x)sin(y)-In(cos(x)) = constant help (formulas) Preview My Answers Submit Answers You have attempted this problem 2 times. Your overall recorded score is 0%. You have unlimited attempts remaining.

Explanation / Answer

(tan(x) -5sin(x)sin(y))dx +(5cos(x)cos(y))dy

fx=M=(tan(x) -5sin(x)sin(y)),fy=N=(5cos(x)cos(y))

My=0-5sin(x)cos(y)=-5sin(x)cos(y)

Nx=-5sin(x)cos(y)=-5sin(x)cos(y)

My=Nx

so equation is exact

so lution is of form f(x,y)=c

f(x,y)=Mdx

f(x,y)=(tan(x) -5sin(x)sin(y))dx

f(x,y)=(sin(x)/cos(x)) -5sin(x)sin(y)dx

f(x,y)=-(ln cos(x)) -5(-cos(x))sin(y) +g(y)

f(x,y)=-(ln cos(x)) +(5cos(x)sin(y)) +g(y)

partially differentiate with respect to y

fy=-0 +(5cos(x)cos(y)) +gy(y)

fy=(5cos(x)cos(y)) +gy(y)

we already had fy=N=(5cos(x)cos(y))

so (5cos(x)cos(y)) +gy(y)=(5cos(x)cos(y))

=>gy(y)=0

=>g(y)=k

so f(x,y)=-(ln cos(x)) +(5cos(x)sin(y)) +k

-(ln cos(x)) +(5cos(x)sin(y)) +k =c

-(ln cos(x)) +(5cos(x)sin(y)) =c-k

-(ln cos(x)) +(5cos(x)sin(y)) =C is the solution to given differential equation

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