Give the correct answer and solve the given equation dx + (frac{x}{y} ​− sin y)dy = 0

Clifland 2021-03-05 Answered
Give the correct answer and solve the given equation dx+(xysiny)dy=0
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

brawnyN
Answered 2021-03-06 Author has 91 answers

Multiply this equation by y:
ydx+(xysiny)dy=0
Now we will try to find a function F: RR2R such that
Fx=y
and Fx=xysiny (1)
Start with Fx=y
and integrate with respect to x:
F(x,y)=ydx=xy+C(y)(2)
where C(y) is a constant with respect to x (an integrating constant!). From (2)
Fx=x+C(y)
Using (1) now, we get that
x+C(y)=xysinyC(y)=ysiny
Thus,
C(y)=ysinydy
=ysinydy
=(u=ydu=dy),(dv=sin y dyv=cosy)
=(y(cos y)cos y dy)
=ycosycosydy
=ycosyy+D,
where D is a constant.
Therefore,
F(x,y)=xy+C(y)=xy+ycosysiny+D
Now the solution of the initial differential equation is given by F(x,y)=0,
so xy+ycosysiny=D=C,
where C is some constant.

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more