V1 - Cálculo III - Universo JF - Prof. Lorena

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Questão 1) 
Y + 2 XY = 0; Y = e^-x^2
Y= e^-X^2
Y' = e^u * U'
e^-X^2 * 2X
Y' = - 2X * e^-X^2
- 2X * e^-x^2 + 2X (e^-x2) = 0
- 2X * e^-X^2 + 2X * e^-x^2 = 0
0 = 0
Questão 2)
A) 
Y' = 2X/1+2Y
dy/dx = 2X/1 - 2y
INT(1-2Y dy) = 2 INT(X DX)
Y + 2Y^2/2 = 2X^2/2
Y = Y^2 = X^2 + C
B) 
dy/dx = SEN 5X
U = 5X
DU/5 = dx
INT(dy) = INT(SEN5X dx)
Y = -1/5 COS 5X + C
C)
2Y^2(X+3) dy = xdx
U = X+3
U-3 = X
du = dx
INT(2Y^2 dy) = INT(X/(X+3 dx)
2Y^3/3 = INT(U-3)/U *DU
INT(u/u*du - 3INT(du/u)
2Y^3/3 = u * 3LN|u| + C
2Y^3 = X+3 - 3LN|X+3| + C
D)
Y' - Y = 3e^4X
u = 3X
du/3 = X
u(Y'-Y) = u * 3e^4X
u(X) = e^INT(-1dx) = e^-X
INT(uY) = INT(u * 3e^4X)
e^X * Y = 3 INT(e^-X * e^4X)
e^X * Y = 3 INT(e^3X dx)
e^-X * Y = 3*1/3 e^3X + C
Y = e^3X/e^-X + c/e-x
Y = e^4X + e^XC
E)
dy/dx + 2Y = X
u = X^2
du = 2X dx
du/2 = X dx
Y' + 2 XY = X
u(Y'+2XY) = uX
u(X) = e^INT(2Xdx) = e^2*(X^2/2) = e^X^2
INT(uY) = INT(uX)
e^X^2 * Y = INT(e^X^2*X dx)
e^X^2 * Y = 1/2 INT (e^u du)
e^X^2 * Y = 1/ * e^X^2 + C
Y = 1/2 *(e^X^2)/(e^X^2) * c/e^X^2
Y = 1/2 + e^-X^2 + c
Questão 3)
dy/dx = (Y+1/X) + Y(2) = 1
Xdy = Y = 1 dx
INT(dy/Y+1) = INT(dx/x)
LN|Y+1| = LN|x| + C
LOG(e^|Y+1|) = LOG (e^|X|) * e^C
Y + 1 = X * e^C
Y = X^e^c - 1
Y = X * 1 - 1
Y= X - 1
Y = 1 X = 2
1 = 2 * e^C - 1
1+1 = 2* e^C
2/ = e^C
1 = e^C