subject
Mathematics, 22.06.2019 21:00 luzcastellanos556

Find the general solution of the given differential equation. x dy dx − y = x2 sin(x) give the largest interval over which the general solution is defined. (think about the implications of any singular points. enter your answer using interval notation.

Answers

ansver
Answer from: katwilf1771

x\dfrac{\mathrm dy}{\mathrm dx}-y=x^2\sin x

Divide both sides by x^2. In doing so, we force any possible solutions to exist on either (-\infty,0) or \boxed{(0,\infty)} (the "positive" interval in such a situation is usually taken over the "negative" one) because x cannot be 0 in order for us to do this.

\dfrac1x\dfrac{\mathrm dy}{\mathrm dx}-\dfrac1{x^2}y=\sin x

Condense the left side as the derivative of a product, then integrate both sides and solve for y:

\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac yx\right]=\sin x

\dfrac yx=\displaystyle\int\sin x\,\mathrm dx

\boxed{y=Cx-x\cos x}

ansver
Answer from: marykm03p3sd80

x\,\dfrac{\mathrm dy}{\mathrm dx}-y=x^2\sin x\implies\dfrac1x\,\dfrac{\mathrm dy}{\mathrm dx}-\dfrac1{x^2}y=\sin x

Note that in order to do this division, we cannot allow x=0. This means the largest interval on which a solution can exist is either (0,\infty) or (-\infty,0).

If y(x) is a solution to the ODE, then any term that vanishes as x\to\infty (or -\infty, depending on which interval above is used) is a transient term.

Solve the ODE:

\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac yx\right]=\sin x\implies\dfrac yx=C-\cos x\implies y=Cx-x\cos x

As x\to\infty, \cos x will oscillate between -1 and 1, so x\cos x will oscillate between -\infty and \infty, so the limit of y(x) as x\to\infty does not exists. There are no transient terms.

ansver
Answer from: baileymtamayo

Divide both sides by x^2 - note that this means we can't have x=0:

x\dfrac{\mathrm dy}{\mathrm dx}-y=x^2\sin x

\dfrac1x\dfrac{\mathrm dy}{\mathrm dx}-\dfrac1{x^2}y=\sin x

Then the left side reduces to the derivative of a product,

\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1xy\right]=\sin x

\dfrac1xy=\displaystyle\sin x\,\mathrm dx=\cos x+C

y=x\cos x+Cx

This solution is continuous everywhere, but accounting for the singular point x=0, the largest interval over which it is defined would be (0,\infty) or (-\infty,0).

Other questions on the subject: Mathematics

image
Mathematics, 21.06.2019 19:00, Marleneg
An energy drink company claims that its product increases students' memory levels. to support its claims, the company issues advertisements claiming that 8 out of 10 people (chosen randomly from across the country) who tried their product reported improved memory. the missing component in this study is a .
Answers: 1
image
Mathematics, 21.06.2019 19:30, christylivingsowzxa2
Use the si prefixes to convert these hypothet- ical units of measure into appropriate quanti- ties. express 49 rations in dekarations. answer in units of dekaration. 004 (part 2 of 5) 10.0 points express 7576 mockingbirds in kilomocking- birds. answer in units of kmockingbird. 005 (part 3 of 5) 10.0 points express 10−4 phones in microphones. answer in units of μphone. 006 (part 4 of 5) 10.0 points express 10−9 goats in nanogoats. answer in units of ngoat. 007 (part 5 of 5) 10.0 points express 1017 miners in examiners. answer in units of eminer.
Answers: 3
image
Mathematics, 21.06.2019 20:00, naomicervero
Abook cover is 7 2/3 inches by 9 5/9 inches. what is the total distance around the cover of the book
Answers: 2
image
Mathematics, 21.06.2019 20:50, emmamood
To solve the system of linear equations 8x+5y = 10 and 6x+y=-2 by using the linear combination method, amos decidedthat he should first multiply the second equation by –5 and then add the two equations together to eliminate the y-terms. hiscalculations are as shown. amos's solution is (2,-14). what did he do wrong? ​
Answers: 3
You know the right answer?
Find the general solution of the given differential equation. x dy dx − y = x2 sin(x) give the large...

Questions in other subjects:

Questions on the website: 13539181