y′=x−2ycot⁡2x

Helen Lewis

Helen Lewis

Answered

2021-12-29

y=x2ycot2x

Answer & Explanation

Piosellisf

Piosellisf

Expert

2021-12-30Added 40 answers

Step 1
Given that the equation y=x2ycot2x
y+(2cot2x)y=x
This is a linear differential equation in y.
dydx+Py=Q
Here, P=2cot2x,Q=x
Pdx=2cot2xdx=2[logsin2x2]=logsin2x
Thus, integral factor, I.F.=eP dx=elogsin2x=sin2x
Step 2
Hence the solution is
ye p dx=Qe p dxdx+C
y(sin2x)=x(sin2x)dx+C
y(sin2x)=[x(cos2x2)+(1)(sin2x4)]+C
y(sin2x)=[xcos2x2sin2x4]+C
ysin2x=14(2xcos2xsin2x)+C is the required solution.
ol3i4c5s4hr

ol3i4c5s4hr

Expert

2021-12-31Added 48 answers

y=x2ycot(2x)
y+2ycot(2x)=x
By inspection, the equation is a linear DE in the form of:
y+yP(x)=Q(x)
where: P(x)=2cot(2x) and Q(x)=x
The integrating factor, i.f. is:
i.f.=eP(x)dx
i.f=e2cot(2x)dx
Note: cotu du=ln(sinu)+C
i.f.=eln(sin(2x))
Note: elnu=u
i.f=sin(2x)
Substituting the i.f., we get:
ysin(2x)=×sin(2x)dx+C
For ×sin(2x)dx, use integration by parts.
In integration by parts, choose u in this order: LIATE
Logarithmic
Inverse
Algebraic
Trigonometric
Exponential
×sin(2x)dx
Using integration by parts:
Let: u=x
du=dx
dv=sin(2x)dx
v=12cos(2x)
udv=uvvdu
×sin(2x)dx=12×cos(2x)12cos(2x)dx
×sin(2x)dx=12×cos(2x)+14sin(2x)
Substituting the value of ×sin(2x)dx, we get:
Vasquez

Vasquez

Expert

2022-01-09Added 457 answers

Simplifying y=x+-2y * cot * 2x Reorder the terms for easier multiplication: y=x+-2 * 2y * cot * x Multiply -2 * 2 Multiply y * cot Multiply cot y * x Reorder the terms: y=-4 cot xy+x Solving y=-4 cot xy+x Solving for variable "y". Move all terms containing y to the left, all other terms to the right. Add "4 cot xy" to each side of the equation. 4 cot xy+y=-4 cot xy+4 cot xy+x Combine like terms: -4 cot xy+4 cot xy=0 4 cot xy+y=0+x 4 cot xy+y=x Reorder the terms: 4 cot xy+-1x+y=x+-1x Combine like terms: x+-1x= 4 cot xy+-1x+y=0 The solution to this equation could not be determined.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?