Ecuaciones Diferenciales

Unidad 1



2.- y’’+4y=0……(1) y= A cos 2x+ B sen 2x……(2)

Comprobación.

y’= -2 A sen 2x+ 2 B cos 2x….(3)

y’’= - 4 A cos 2x - 4 B sen 2x….(4)

Sustituir 4 y 2 en 1.

-4 A cos 2x – 4 B sen 2x + 4 (A cos 2x + B sen 2x)

-4 A cos2x – 4 B sen 2x + 4 A cos 2x + 4 B sen 2x = 0

Por lo tanto es solución 0 = 0





4. - y’’’= 6…. (1)y=x3 + ax2 + bx + c…….. (2)

Comprobación. Sustituir (5) en (1)

y’= 3x2 + 2ax + b…..(3) 6 = 6



y’’= 6x + 2a………(4)

y’’’= 6…………(5)







6.- y’= sen x…. (1) (usando el método de separación de variables)Comprobación.

[pic]= sen x [pic] y= - cos x +c

dy= sen x dx….(2) y= - cos x +c y’= - (cos x+c)

y’= - (-sen x)

Sustituir (3) en (1)y’= sen x…..(3)

sen x = sen x



10.- y’ + y = 0…. (1) y= ce-x …..(2)

Comprobación sustituimos 3 y 2 en 1

y= ce-x - ce-x + ce-x = 0

y’= - ce-x …..(3) 0 = 0



12.- xy’ – 4y = 0…. (1)y = cx4…..(2)

Comprobación sustituimos 4 y 2 en 1

y = cx4…..(3) 4cx3 – 4(cx4) = 0

y´= 4cx3….(4)

y’’=12 cx2….(5)

y’’’= 24cx…(6)



16.- y’=1; y= y (7) = 0; y= x+c comprobación

0 = 7 + c y = x + c

c = -7y’= 1





18.- [pic]

[pic] [pic] =[pic]

y[pic]=[pic]

y[pic]=2[pic]

y=[pic]+[pic]

y= 2 + C[pic]



Pagina 29.

6.- y’= cos y

[pic]

[pic]

[pic]= [pic]

In /sec y + tan y /= x + C

sec y + tan y= [pic]= [pic]

sec y + tan y=[pic] solución general



8.- [pic]

[pic]

[pic]

y[pic] [pic]u=x dv=[pic]
du=dx v=-[pic] [pic]

y[pic]= [pic] [pic] [pic] dx+C

y[pic]= - [pic] [pic] - [pic] [pic] + C

y= C[pic]- [pic] x - [pic] solución general

10.- [pic]

seau=[pic]+[pic]

u´=2x+2y y´

y=[pic]

u´- 2x =2 yu

u´-2yu= 2x [pic] = [pic]



u´[pic] = 2 [pic] dx + C

u´[pic] =- x u´[pic] + [pic] [pic] + C

[pic]= [pic] - [pic] + C

[pic]= - [pic] - + [pic] + C [pic] solución general



12.- 4y´+ y = 0

4 [pic]= -y

4 [pic]=- dx

4 [pic]= - [pic]

4 /In y / =-x + C/In [pic]/ =-x + C

[pic]=[pic] = [pic] = [pic]

[pic]

y=[pic] y=[pic] solucion general.



14.- y´+[pic]=0

[pic]

[pic]= - dx

[pic] = - [pic]

- [pic]= -x +C por (-1)

[pic]

1= Cx+ Cy

y= [pic]



Pagina 35.

6. - xy´+by=o

x[pic]+=-by

[pic] = - [pic]

[pic] = - b [pic]

In /y/ =-b /In x/ + In /c/

In /y/=In /[pic] / + In /c/In /y/ = In / C [pic]

y= C[pic] solucion general



8. - (x + 2) y´- xy = 0

(x + 2) [pic] - xy = o multiplicar por ( [pic] )

( [pic] ) [pic] * [pic] - [pic] = 0

[pic] - [pic] = 0

[pic]- [pic] = 0 [pic] - x -2 = 1 - [pic]

[pic] dx = 0

In /y/ - x + 2 In / x + 2 /= C=

In /y/ + In /x+ 2 /[pic]= x + C

Y(x+2)2...