Ecuaciones De Segundo Orden

Ejercicios 2.2
Daniro rojas
Grupo DD

1. dydx = sen 5x

dy = sen 5x dx
dy = sen 5x dx u = 5x du5 = dx
y = 15sen u du
y = -15cosu+c
y = -15cos5x+c
2. dydx=(x + 1)2
dy =(x + 1)2 dx
dy= (x + 1)2 dx u = x + 1 du = dx
y= u2 du
y= 13 u3 + c
y= 13 (x + 1)3 + c
3. dx + ℮3x dy = 0
dy = - ℮-3x dx
dy = - ℮-3x dx u = -3x - du3 = dx
y =13 - ℮u dx
y =-13 ℮u + c

4. dy – (y – 1)2 dx = 0
-(y – 1)2 dx = dy
dx = dy
(y – 1)2
dx = dy (y – 1)2 u = y -1 du = dy
x=du (u)2
x = u-2 du
x = u -1
x = (y –1)-1
x = 1(y –1) - 1(y–1) = x + c

5. x dydx = 4y
x dyy = 4dx
dyy = 4dxx
dyy = 4dxx
lny = 4dxx
lny = 4lnx

6. dydx + 2xy2 = 0
dydx = - 2xy2
dyy2 = -2x dx u = y du = dy
du u2 = -2x dx
u-2du= -2 x22 + c
u-1 = - x2 + c
1u = - x2 + c
1y = - x2 + c

7. dydx = e3x +2y
e-2y dy = e3x dx
e-2y dy = e3x dx u = -2y -du2=dy ; v = 3x dv3=dx
-12 eu du = 13 ev dv
-12 e-2y =13 e3x + c

8. exy dydx = e -y + e -2x-y
ex ydy = (e –y + e -2x-y) dx
-ey – ey dy = (e)-2x(e)x dx
-2eyydy = e-3xdx
-2yey dy = e-3x dx u=y du = dy; v =ey dv= yey uv – v du m = -3x ;-dm3 = dx
-2 yey - ey dy = -13emdm q = y dq = dy
-2 yey - eq dq = -13e-3x + C
-2 yey – ey =-13 e-3x + C
-2 yey – ey + 13e-3x = C

9. y lnx dxdy = (y+1x)2
y lnx dxdy = (y+1)2(x)2
yx2ln x dx = (y)2+2y1 + 12dy
x2 ln x dx = (y)2y+2yy + 1y dy
x2 ln x dx = y + 2 +1y dy
x2 ln x dx = ydy + 2dy +1y dy u= lnx du = 1x dx dv = x2 dx v = (x)33
lnx (x)33 - (x)33 1x dx = (y)22 +2y + lny
lnx (x)33 - 13 x3 x-1 dx = (y)22 + 2y + lny
(x)33 lnx - 13 x2dx = (y)22 + 2y + lny
(x)33 lnx - 19 x3 + C = (y)22 + 2y + lny

10. dydx=(2y+34x+5)2
dy(2y+3)2 = dx(4x+5)2
dy(2y+3)2 =dx(4x+5)2 u = 2y+5 du2 = dy w = 4x + 5 dw 4= dx
12 du(u)2 = 14dw(w)2
12 (u)-2 = 14(w)-2
12 (u)-1 -1= 14 (w)-1-1
12 (2y+3) = -14(4x+5)
42 (2y+3) = 1(4x+5)
2 (2y+3) = 1(4x+5) + C...