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

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

Question
Integrals
asked 2021-03-05
Give the correct answer and solve the given equation \(dx + (\frac{x}{y} ​− \sin y)dy = 0\)

Answers (1)

2021-03-06
Multiply this equation by y:
\(y dx + (x ​− y \sin y)dy = 0\)
Now we will try to find a function F: \(RR^{2} \rightarrow R\) such that
\(\frac{\partial F}{\partial x}=y\)
and \(\frac{\partial F}{\partial x} = x - y \sin y\) (1)
Start with \(\frac{\partial F}{\partial x} = y\)
and integrate with respect to x:
\(F(x, y) = \int y dx = xy + C(y) (2)\)
where C(y) is a constant with respect to x (an integrating constant!). From (2)
\(\frac{\partial F}{\partial x} = x + C'(y)\)
Using (1) now, we get that
\(x + C'(y) = x - y \sin y \Rightarrow C'(y) = -y \sin y\)
Thus,
\(C(y) = \int -y \sin y dy\)
\(= - \int y \sin y dy\)
\(= {(u = y \Rightarrow du = dy),(dv = \sun\ y\ dy \Rightarrow v = -\cos y)}\)
\(= -(y(-\cos\ y) - \int - \cos\ y\ dy)\)
\(= y \cos y - \int \cos y dy\)
\(= y \cos y - \sim y + D,\)
where D is a constant.
Therefore,
\(F(x, y) = xy + C(y) = xy + y \cos y - \sin y + D\)
Now the solution of the initial differential equation is given by \(F(x, y) = 0\),
so \(xy + y \cos y - \sin y = - D = C,\)
where C is some constant.
0

Relevant Questions

asked 2021-03-06

Give the correct answer and solve the given equation \([x-y \arctan(\frac{y}{x})]dx+x \arctan (\frac{y}{x})dy=0\)

asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2020-11-02
Give the correct answer and solve the given equation \(\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{{y}^{2}-{1}}}{{{x}^{2}-{1}}},{y}{\left({2}\right)}={2}\)
asked 2021-01-05
Give the correct answer and solve the given equation
\(\displaystyle{\left({x}+{y}\right)}{\left.{d}{x}\right.}+{\left({x}-{y}\right)}{\left.{d}{y}\right.}={0}\)
asked 2020-11-24
Give the correct answer and solve the given equation \(\displaystyle{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{\left({3}{x}+{y}\right)}{\left.{d}{y}\right.}={0},\text{when}\ {x}={3},{y}=-{2}\)
asked 2021-03-22
Solve the equation:
\(\displaystyle{\left({x}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}{\left({y}^{{2}}+{1}\right)}\)
asked 2021-05-10
Solve the equation:
\(\displaystyle{\left({a}-{x}\right)}{\left.{d}{y}\right.}+{\left({a}+{y}\right)}{\left.{d}{x}\right.}={0}\)
asked 2020-11-11
Give the correct answer and solve the given equation Evaluate \(\displaystyle\int{x}^{3}{\left({\sqrt[{3}]{{{1}-{x}^{2}}}}\right)}{\left.{d}{x}\right.}\)
asked 2020-11-01
Give the correct answer and solve the given equation
Let \(\displaystyle{p}{\left({x}\right)}={2}+{x}{\quad\text{and}\quad}{q}{\left({x}\right)}={x}\). Using the inner product \(\langle\ p,\ q\rangle=\int_{-1}^{1}pqdx\) find all polynomials \(\displaystyle{r}{\left({x}\right)}={a}+{b}{x}\in{P}{1}{\left({R}\right)}{P}\)
(R) such that {p(x), q(x), r(x)} is an orthogonal set.
asked 2021-03-20
The graph of y = f(x) contains the point (0,2), \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}\), and f(x) is greater than 0 for all x, then f(x)=
A) \(\displaystyle{3}+{e}^{{-{x}^{{2}}}}\)
B) \(\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}\)
C) \(\displaystyle{1}+{e}^{{-{x}}}\)
D) \(\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}\)
E) \(\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}\)
...