Differential Equations Answers

Show that the equations (y - z)p+ (z-x)q = x - y and z- - px - qy = 0 are compatible

Show that the equations (y - z)p+ (z-x)q = x - y and z- - px - qy = 0 are compatible.

{Fe} Obtain two linearly independence solution valid near the origin for the following equation

𝑥𝑦"+ 3𝑥𝑦′ + ( 1+4𝑥2) 𝑦=0

<e> (d^3-7dd'^2-6dd'3)z=e^2x+y +cos(x+y)+x^2y

<e> Q.(a): Construct the solution of the heat equation using separation of variables method:

u_xx=4u_t , 0<x<40, t>0

u(0, t)=0, u(40, t)=0, t>0

u(x,0)= x, 0 less than or equal x less than or equal 20, u(x,0)=40-x, 20 less than or equal x less than or equal 40

(b): Find the steady-state solution of the heat equation u_xx=4u_t , 0<x<40, t>0 that satisfies the Bcs:

u(0, t)=10 , u(40, t)=40

(c): Plot u versus x for several values of t.

(d): Plot u versus t for several values of x.

(e): Determine how much time must elapse before the temperature at x=40 comes within 1 centigrade of its steady state.smi

<e> a) A rod of length 𝐿 has its ends 𝐴 and 𝐵 maintained at 20 °𝐶 and 40 °𝐶 respectively

until steady state condition prevail. The temperature at 𝐴 is suddenly raised to

50 °𝐶 while that at 𝐵 is lowered to 10 °𝐶 and maintained thereafter. Find the

subsequent temperature distribution of the rod.

b) Solve (𝑥

2 − 𝑦

2 − 𝑧

2

)𝑝 + (2𝑥𝑦) 𝑞 = 2𝑥𝑧 where 𝑝 =

𝜕𝑧

𝜕𝑥 , 𝑞 =

𝜕𝑧

𝜕𝑦

<e> A uniform string of line density rho stretched to tension rho.a^2 ,executes small transverse vibration in a plane through the undisturbed line of the string. the ends x=0 ; x= l of the string are fixed . Point x=b drawn aside through a small distance d and released at time t=0. Find the transverse displacement of any point of the string at any subsequent time.