Topic: Exact Differential Equations Instructions: N

Helen Lewis

Helen Lewis

Answered question

2021-12-18

Topic:
Exact Differential Equations
Instructions:
Solve the Exact Differential Equations of GIven Problem #2 only, ignore the other given problem.
Note:
Please answer all of the problems in the attached photo. I will rate you with “like/upvote” after; if you answer all of the problems. If not, I will rate you “unlike/downvote.” Kindly show the complete step-by-step solution.
1. (xexy+exy)dx+(8+xex)dy=0;y(0)=4
2. (x+yy2+1)dy+(arctany+x)dx=0

Answer & Explanation

zesponderyd

zesponderyd

Beginner2021-12-19Added 41 answers

(2) Given differential equation
(x+yy2+1)dy+(arctany+x)dx=0
(tan1(y)+x)dx+(x+yy2+1)dy=0
Comparing with Mdx+Ndy=0
M=tan1(y)+x
My=11+y2+0
My=11+y2
N=x+yy2+1=xy2+1+yy2+1
Nx=1y2+1+0
Nx=11+y2
My=Nx
Given differential equation is exact differential equation.
Silution of given differential equation
ndx+N(terms free from x)dy=c (1)
N(terms of N-free from x)=yy2+1dy=122yy2+1
from 1,2 and 3 =12ln(y2+1)(3)
[f(x)f(x)dx=ln(f(x))+c
xtan1(y)+x22+12ln(y2+1)=c
Shawn Kim

Shawn Kim

Beginner2021-12-20Added 25 answers

1. (xexy+exy)dx+(2+xex)dy=0,y(0)=4
comparing with Mdx+Ndy=0
M=(xex+ex)y and N=(8+xex)
Now My=xex+ex and Nx=0+ex+xex
=xex+ex
Hence My=Nx. Given differential equation is exact.
Hence solution is given by
Mdx+(The term in N free from x)dy=c
(x+1)exydx+8dy=c
y[(x+1)ex(ddx(x+1)exdx)dx]+8y=c
y[(x+1)exexdx]+8y=c
y[(x+1)exex]+8y=c
yex(x+11)+8y=exyex+8y=c
Given initial condition as y(0)=1 at x=0,y=1
0+8[a)=cc=32
Hence xyex+8y=32xyex+8y+32=0 be the solution of given differential equation.
RizerMix

RizerMix

Expert2021-12-29Added 656 answers

1) (xexy+exy)dx+(8+xex)dy=0
y(0)=4
Compare with MdxNdy=0
M=xexy+exy;N=e+xex
My=xx+ex Nx=1.ex+xex
=ex(x+1) =ex(x+1)
So, My=Nx
So, given Eq is exact diff. eq. whose
Mdx+Ndy=c
y=c terms not
Cont. x
(xexy+exy)dx+(8+xex)dy=c
yxexdx+yexdx+8y=c
y[xedxx[ddx(x)exdx]dx+yex+8y=c
y[xex1exdx]+yex+8y=c
y[xexex]+yex+8y=c
xyexexy+yex+8y=c
xyex+8y=c
put x=0 and y=4
px(m)e+8(4)=c
c=32
So xyex+8y=32
2) (x+yy2+1dy+(tan1y+x)dx=0
Sol: (tan1y+x)dx+(x+yy2+1)dy=0
Compare Mdx+Ndy=0
M=y+x:N=x+yy2+1
My=1y2+1;Nx=1y2+1
My=Mx
So, exact diff. Equation
Mdx+Ndy=c
y=c
(tan1y+x)dx+(x+yy2+1)dy=c
xtan1y+x22+yy2+1dy=c
Let y2+1=t

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