INSTITUTO POLITECNICO NACIONAL

UNIDAD PROFESIONAL INTERDISCIPLINARIA DE INGENIERIACAMPUS GUANAJUATO

Ecuaciones Diferenciales : Tarea1

JANAI ARRIAGA Disponible desde : 08/16/2012 at 12:01pm CDT.Fecha de cierre : 08/24/2012 at 11:55pm CDT.

INSTRUCCIONES: Resuelve los siguientes problemas. No olvides justificar tus respuestas. Exito!.

1. (1 pt) It can be helpful to classify a differential equation,so that we can predict the techniques that might help us to find afunction which solves the equation. Two classifications are theorder of the equation (what is the highest number of deriva-tives involved) and whether or not the equation is linear .Linearity is important because the structure of the the family ofsolutions to a linear equation is fairly simple. Linear equationscan usually be solved completely and explicitly.

Determine whether or not each equation is linear:

? 1. y y+ y2 = 0? 2. t2

d2ydt2

+ tdydt

+2y= sin t

? 3. y y+ t2 = 0? 4.

dydt

+ ty2 = 0

2. (1 pt) Match each differential equation to a function whichis a solution.FUNCTIONSA. y= 3x+ x2,B. y= e5x,C. y= sin(x),D. y= x

12 ,

E. y= 4exp(5x),DIFFERENTIAL EQUATIONS

1. xy y= x22. y+ y= 03. y+11y+30y= 04. 2x2y+3xy = y

3. (1 pt) It is easy to check that for any value of c, the func-tion

y= ce2x+ ex

is solution of equation

y+2y= ex.

Find the value of c for which the solution satisfies the initialcondition y(5) = 4.c=

4. (1 pt) It is easy to check that for any value of c, the func-tion

y= x2 +cx2

is solution of equation

xy+2y= 4x2, (x> 0).

Find the value of c for which the solution satisfies the initialcondition y(1) = 9.c=

5. (1 pt) Which of the following functions are solutions ofthe differential equation y6y+9y= 0?

A. y(x) = xe3x B. y(x) = 3x C. y(x) = 3xe3x D. y(x) = x2e3x E. y(x) = 0 F. y(x) = e3x G. y(x) = e3x

6. (1 pt) Find the two values of k for which

y(x) = ekx

is a solution of the differential equation

y21y+104y= 0.

smaller value =larger value =

7. (1 pt) Solve the separable differential equation

dydx

=0.5cos(y)

,

and find the particular solution satisfying the initial condition

y(0) =pi3.

y(x) = .

1

8. (1 pt) Solve the separable differential equation

dxdt

= x2 +1

16,

and find the particular solution satisfying the initial condition

x(0) = 7.

x(t) = .

9. (1 pt) Find k such that x(t) = 16t is a solution of the dif-

ferential equationdxdt

= kx.k = .

10. (1 pt) Use the mixed partials check to see if the follow-ing differential equation is exact.If it is exact find a function F(x,y) whose differential, dF(x,y)is the left hand side of the differential equation. That is, levelcurves F(x,y) =C are solutions to the differential equation:

(4x3 +2y)dx+(4x2y2)dy= 0First:My(x,y) = , and Nx(x,y) = .

If the equation is not exact, enter not exact, otherwise enterin F(x,y) here

11. (1 pt) Use the mixed partials check to see if the follow-ing differential equation is exact.If it is exact find a function F(x,y) whose differential, dF(x,y)is the left hand side of the differential equation.

That is, level curves F(x,y) =C are solutions to the differen-tial equation

(2ex sin(y)2y)dx+(2x2ex cos(y))dy= 0First:My(x,y) = , and Nx(x,y) = .

If the equation is not exact, enter not exact, otherwise enterin F(x,y) here

12. (1 pt) The differential equation

y+3y5 =(y4 +4x

)y

can be written in differential form:

M(x,y)dx+N(x,y)dy= 0

whereM(x,y) = , and N(x,y) = .

The term M(x,y)dx+N(x,y)dy becomes an exact differen-tial if the left hand side above is divided by y5. Integratingthat new equation, the solution of the differential equation is

=C.

13. (1 pt) A Bernoulli differential equation is one of the formdydx

+P(x)y= Q(x)yn.

Observe that, if n = 0 or 1, the Bernoulli equation is linear.For other values of n, the substitution u = y1n transforms theBernoulli equation into the linear equation

dudx

+(1n)P(x)u= (1n)Q(x).

Use an appropriate substitution to solve the equation

xy+ y=5xy2,

and find the solution that satisfies y(1) = 4.

y(x) = .

14. (1 pt) Find the particular solution of the differential equa-tion

dydx

+ ycos(x) = 3cos(x)

satisfying the initial condition y(0) = 5.Answer: y(x)= .

15. (1 pt) Find the function satisfying the differential equa-tion

y3y= 4e5tand y(0) = 0.

y= .

16. (1 pt) Solve the initial value problem

10(t+1)dydt7y= 21t,

for t >1 with y(0) = 16.y= .

17. (1 pt) A. Let g(t) be the solution of the initial valueproblem

4tdydt

+ y= 0, t > 0,

with g(1) = 1.Find g(t).g(t) = .

B. Let f (t) be the solution of the initial value problem

4tdydt

+ y= t3

with f (0) = 0.Find f (t).

2

f (t) = .C. Find a constant c so that

k(t) = f (t)+ cg(t)

solves the differential equation in part B and k(1) = 19.c= .

18. (1 pt) Solve the initial value problemdxdt

+2x= cos(4t)

with x(0) =3.x(t) = .

19. (1 pt) Find the function satisfying the differential equa-tion

f (t) f (t) = 9tand the condition f (2) =4.f (t) = .

20. (1 pt) Find u from the differential equation and initialcondition.

dudt

= e1.2t2.3u, u(0) = 3.8.

u= .

Generated by the WeBWorK system cWeBWorK Team, Department of Mathematics, University of Rochester

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