Diferentiai equations La Ecuacion Diferencial x dy/dx = 2y + sinx es: a) Lineal

Question

Diferentiai equations La Ecuacion Diferencial x dy/dx = 2y + sinx es: a) Lineal b) Homogenea c) Separable d) Exacta e) Bernoulli Resueive la Ecuacion Differencial x dy/dx = (x + 1)y^2 a) y = 1/x + lnx + c b) y = 1/x - lnx + c c) y = -c/x + lnx d) y = -c/x - lnx e) y = -1/x + lnx + c Resueive la Ecuacion Diferencial Exacta (siny - ysinx)dx + (cosx + xcosy - y)dy = 0 a) xcosy + xsiny + y^2/2 = c b) xcosy + ysinx - y^2/2 = c c) xsiny + ycosx -y^2/2 = c d) xsiny + ycosx + y^2/2 = c e) ysinx + xcosx + y^2/2 = c

Explanation / Answer

1.

y'' - 4y = 0

Put y = e^mx

y' = m*e^mx

y'' = m^2*e^mx,

m^2*e^mx - 4*e^mx = 0

m^2 - 4 = 0

m^2 = 4

m1 = +2 and m2 = -2, So

y1 = e^2x and y2 = e^-2x

Correct option is C.

2.

x*dy/dx = 2y + sin x

dy/dx = 2y/x + (sin x)/x

dy/dx - 2y/x = (sin x)/x

Above equation is Bernoulli

Correct option is E.

3.

x*dy/dx = (x+1)*y^2

dy/dx = (1 + 1/x)*y^2

dy/y^2 = (1 + 1/x)*dx

-1/y = x + ln x + C

y = -1/(x + ln x + C)

Correct option is E.

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