For each of the following differential equations, determine the general or particular solution: 2xy′+y=y2log⁡x

David Troyer

David Troyer

Answered

2021-12-26

For each of the following differential equations, determine the general or particular solution: 2xy+y=y2logx

Answer & Explanation

Mary Herrera

Mary Herrera

Expert

2021-12-27Added 37 answers

Step 1
Given:
Differential equation
xy+y=y2logx
To find :
General solution of the given Differential equation?
Step 2
Solution :
We have given
xy+y=y2logx
The given DE equation is a Bernoulli equation of the form
dydx+P(x)y=Q(x)yn
And we can solve this kind of equation by substituting
v=y1n
So we have the equation xy+y=y2logx
and here the value of n is 22
So we will substitute
v=y12
v=1y
Now differentiate with respect to x
dvdx=1y2dydx
dydx=y2dvdx
Now we can write the given equation as
xdydx+y=y2logx
dydx+yx=y2logxx
Now put the value of dydx=y2dvdx
y2dvdx+yx=y2logxx
divide both the sides by y2
dvdx1xy=logxx
Now put v=1y
dvdxvx=logxx
Step 3
We got the equation
dvdxvx=logxx
Now we will find out the integrating factor for the above equation
P(x)=1x 
esfloravaou

esfloravaou

Expert

2021-12-28Added 43 answers

Rewrite Bernoulli Equation in the appropriate form:
xy+y=y2lnx
dydx+1xy=lnxxy2
Use a substitution:
v=y12=y1
dvdx=y2dydx
=y2(1xy+lnxxy2)
=1xy1lnxx
=1xvlnxx
dvdx1xv=lnxx
Find integrating factor:
μ=e1xdx=elnx=1x
1xdvdx1x2v=lnxx2
Solve for v:
ddx(vx)=lnxx2
d(vx)=lnxx2dx
Integrate by parts:
vx=lnxx2dx
=1xlnx1x2dx
=1xlnx+1x+C
v=Cx+lnx+1
Substitute back and solve for y:
y1=Cx+lnx+1
karton

karton

Expert

2022-01-09Added 439 answers

You differentiate implicitly, y+xdy/dx+dy/dx=2ydy/dx×1/x
Rearranging
xdy/dxdy/dx+2ydy/xdx=y
Factorising dydx
dydx=yx1+2yx

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