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

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

asked 2021-03-06

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

Answers (1)


We will first write this equation as
\(x \arctan (\frac{y}{x}) \cdot \frac{dy}{dx} = y \arctan (\frac{y}{x}) - x\) (1)
Notice that \(\arctan (\frac{y}{x}) = 0\)
means that \(\frac{y}{x} = 0,\ so\ y = 0.\) However, this is
not a solution of the given equation.
Thus, \(\arctan (\frac{y}{x}) \neq 0,\) so we can divide the equation (1)
by \(x \arctan (\frac{y}{x})\)
\(\frac{dy}{dx}=\frac{y}{x}-\frac{1}{\arctan(\frac{y}{x})}\) (2)
Make the substitution \(u = \frac{y}{x}\ or\ y = ux.\) Then
\(\frac{dy}{dx} = \frac{d}{dx} (ux) = x \frac{dy}{dx} + u\) (3)
(use the Chain Rule). Furthermore, (2) becomes
\(\frac{dy}{dx} = u - \frac{1}{\arctan(u)}\) (4)
Combining (3) and (4), we get \(x \frac{du}{dx} + u = u - \frac{1}{\arctan(u)} \Rightarrow x \frac{du}{dx} = - \frac{1}{\arctan(u)}\)
Notice that this is a separable equation! It can be written as
\(\arctan(u) du = - \frac{1}{x} dx\)
Integrate both sides:
\(\int \arctan(u) du = - \int \frac{dx}{d}\) (5)
\(- \int \frac{dx}{d} = - \ln |x| + C_{2},\)
where \(C_{2}\) is some constant.
For the other integral, we use the integration by parts:
\(\int \arctan(u) du = \{(t = \arctan(u) dv = du), (dt = \frac{du}{1+u^{2}} v = u)\}\)
\(= u \arctan(u) - \int \frac{udu}{1+u^{2}} du\)
\(= u \arctan(u) - \frac{1}{2} \ln |1 + u^{2}| + C_{1}\)
\(= u \arctan(u) - \ln \sqrt{1+u^{2}} + C_{1}\)
where \(C_{1}\) is some constant.
(In last equality we need used that \(1 + u^{2} > 0,\ so\ |1 + u^{2}| = 1 + u^{2},\)
and the property of the logarithmic function: \(x \ln y = \ln y^{x}.)\)
Finnaly. (5) becomes
\(= u \arctan(u) - \ln \sqrt{1+u^{2}} = - \ln |x| + C,\)
where \(C = C_{2} - C_{1}\)


Relevant Questions

asked 2021-03-09

Show by substitution that \(u(x,t)=\cos(απx)e^−α^2π^2t\) is a solution of the heat equation \(ut=uxx\) on any interval [0, L].

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 2021-03-22
Solve the equation:
asked 2021-03-05
Give the correct answer and solve the given equation \(dx + (\frac{x}{y} ​− \sin y)dy = 0\)
asked 2021-05-10
Solve the equation:
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) \(\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}\)
asked 2021-02-27
Solve the following differential equations:
asked 2021-06-08
Find the first and second derivatives.
asked 2021-01-06

Give the correct answer and solve given
a. Test for a difference in the means in the two populations using an \([\alpha={.05}{t}-{t}{e}{s}{t}.]\)
b. Place a 95% confidence interval on the difference in the means of the two populations.
c. Compare the inferences obtained from the results from the Wilcoxon rank sum test and the -test.
d. Which inferences appear to be more valid, inferences on the means or the medians?

asked 2021-03-07

Solve differential equation \(\frac{\cos^2y}{4x+2}dy= \frac{(\cos y+\sin y)^2}{\sqrt{x^2+x+3}}dx\)