(5x + 6y) dx + (8y^3 - 6x) dy = 0 and x dy/dx = 2xe^x - y + 6x^2

Question

determine whether the given differential equation is exact. If it is exact, solve it.

Answer 1

→ dy/dx = 5x+6y/6x-8y^3 → (1)

Answer 2

x → 1/t : y(x) → y(t)

Answer 3

dy/dx=dy/dt (dt/dx)

Answer 4

= -t^2 dy/dt → (2)

Answer 5

(1),(2) → -t^2•dy/dt

Answer 6

=(5/t)+6y/(6/t)-8y^3

Answer 7

→ dy/dt=5+6yt/8ty^5-6t^2

Answer 8

(5+6yt)dt+(6t^2-8ty^5)dy=0

Answer 9

P_=5+6yt

Answer 10

Q_=t•(6t-8y^5)

Answer 11

æp/æy=6t

Answer 12

æQ/æt=12t-8y^5

Answer 13

lnIf={6t-(12t-8y^5)/t(6t-8y^5) dt

Answer 14

={-dt/t

Answer 15

If= 1/t , If≠ 1

Answer 16

→ Ē u(y,t):

Answer 17

du_= Utdt+Uy dy=0

Answer 18

→ u= CONST.

Answer 19

æu/æy_=I,fQ

Answer 20

=6t-8y^5

Answer 21

→ u_=6ty-40y^4+w(t) →(3)

Answer 22

æu/æt=6y+w‘(t)_= I,f P

Answer 23

→w‘(t)=5/t

Answer 24

W(t)= lnt^5 → (4)

Answer 25

(3),(4)→ u_=6ty-40y^4+5 lnt=c → (5)

Answer 26

t → 1/x :

Answer 27

6y/x-40y^4-5 lnt=c → (6)

Answer 28

6y/x-40y^4-5 lnx=c → (6)

Answer 29

Answer for x dy/dx=2xe^x-y+6x^2

Answer 30

(Y-6x^2-2xe^x) dx+xdy=0

Answer 31

P_=y-6x^2-2xe^x

Answer 32

Q_=x and æp/æy=1

Answer 33

= æQ/æx therefore the given is exact:

Answer 34

du_=Pdx+Qdy=0

Answer 35

U(x,y)=c

Answer 36

æu/æy=x

Answer 37

U=xy+w(x)

Answer 38

And æu/æx=y+w'(x)_= y-6x^2-2xe^x

Answer 39

Then w(x)

Answer 40

{6x^2+2xe^x) dx

Answer 41

=-2x^3-2{xd(e^x)

Answer 42

=-2x^3-2(xe^x-e^x)

Answer 43

W(x)=-2x^3-2xe^x+2e^x

Answer 44

Therefore xy=2x^3+2xe^x-2e^x+c

Answer 45

Y=2x^2+2•(1-1/x)e^x+c/x