Dudas Homogenes

(x-y)(4x+y)dx+x(5x)dy=0

(4x2+xy-4xy-y2)dx + (5x2-xy)dy=o

[4x2-3xy-y2]dx + (5x2-xy)=0

4x2t2-3xtyt- 42t2 5x2t2-ytyt

[4x2-3xy-y2] e2 t2 [5x2 – xy]

E.D.h

Y=ux x=uy

Dy=udx+xdu dx=udy+ydu

[4x2-3xux-u2x2] dx + (5x2-xux) (udx+xdu)=0

4-3 u – u2] x2 + (5+u) x2 udx + (5-u) x3du=0

[4-3u-u2+5u-u2] x2 dx= - (5-u)x2du

2[2+u-u2] x2 dx = (u-5) x3 du

2x2 /x3 dx = (u-5)/2+u-u2 du

2dx/x ∫▒〖5-u 〗 du= ∫ 5- u/ u-1/2-9/4= ∫ 5-[w+1/2] / w2- 9/4 du

W=u- ½ dw= du u = w+ ½

2(∫ 9)/2- w /w2- (3/2)2 dw=(∫ A)/(w2 )-(3/2)+ B w

W= -3⁄2 -3B=6 B=-2 2Inx =1

W = 3⁄2 3a= 3 A=1

2Inx= in(w- 3⁄(2)) = 2In |w+3⁄(2| )+ Inc

2Inx = In| u-(-1)⁄2- 3⁄(2)) - 2In (u-1⁄(2+ 3⁄(2)+Inc ))

Inx2 = In (u-29-2 In (u+1)+Inc Sea M(x,y) dx + n(x,y) dy = 0

Si 2m/(2y ) = 2n/2n E. D. E

Inx2 = (2x+y) dx –( x+gy) dy =0

2/2y (2x+y) 2/(2x ) - (x+6y)

[x-y Iny+yInx] dx+x(Iny-Inx) dy =0

{x-y [Iny + inx]} dx + x (Iny-inx) dy=0 (2x+y) dx + (x+6y) dy = 0

(x-y In y/(a ) ) dx + x In y/a dy =0 2/2y ( 2x+y) d/dx ( x+6y) 1=1 E.D exacta

(x-y In y/(x )) t t (x In y/x 2f/2x = 2x+y 2f/2x = x+6y

E:D homogenea grado 1 ∫▒2f ∫▒〖(2x+y)dy 〗

Y= ux x=uy f= 2 ∫▒〖xdx+y ∫▒dx〗

Dy=udx+ xdu dx= udy+ ydu f=x2 +xy+g(y)

( In u In u) x dx + u In uxdx + x2 in udu = 0 2f/2y = d/2y [x2 + xy+g (y)]

|1| xdx= x2 In udu 2f/2y = 2y/2y - g (y)

dx= in udu 2f/2y = x+g (y)

2f/2y = x + 6y

dq/dy = 6y g (y) = 3x2 + c

Inx= - [u In u – ∫▒〖du/u u 〗

In x = - u In u - ∫▒〖du/ 〗

In x = u – u In u – u

Y= ux u= y/x

In x = y/(x- - ) x/(x ) In + c

(sen y – sen x) dx + (cosx+xcosy –y) =0 d/dy sen y- y sen x = cosy – sen x

2/2x cos x + x cosy- y = - sen x + cos y E.d exacta

df/(da ) = sen y- y sen x 2f/2y = cos x + x cos y – y

∫▒〖df 〗 = ∫▒〖sen y-y sen x ) dx 〗

F= sen y ∫▒〖sen y ∫▒〖dx-y ∫▒〖sen x dx 〗〗〗

F= sen y x + y cos x + g (y)

2f/2y = x d/2y sen y + cos x 2/2uy + 2/2y g (x)

2f/(2y ) = x cos x + cos x + g (y)

2f/(2y ) = x cos y + cos x + g (x)

2f/(2y ) = cos +xcos y –y

dg/dy = - y ∫▒〖dg= - ∫▒〖y dy 〗〗 g= - y/2 + c

F= x sen y + y cos x – y2 / 2 + c

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