so here we have given that an inhomogeneous equation and we have to find the solution f(t,x) with the help of
DUHAMEL PRINCIPLE WHICH relates the solution of the given equation with auxiliary solution
and we have given already the Auxiliary equation .so here we dont need to find it
we just find the solution f(t,x) with the help of this formula given below
so f(t,x)=
w(t,x,s) ds
=2e37t/8-x/2 sin3x (s.e-37s/8)ds
now we solve with help of integration by parts which is stated below
=2 e37t/8-x/2 sin3x [s{e-37s/8
(-8/37)}+ 8/37
e-37s/8 ds]
=2 e37t/8-x/2 sin3x [s{e-37s/8(-8/37)} + 8/37 {(-8/37)e-37s/8}]
taking limit 0 to t in complete integral where you can see the closed bracket [ ]
=2.e37t/8-x/2 sin3x [(-8/37t.e-37t/8)-(64/1369.e-37t/8+ 64/1369)]
= e-x/2 sin3x [ -16/37 t - 128/1369 + 128/1369 e37t/8]
= 16/1369 .e-x/2 sin3x [-37t-8+8.e37t/8]
so option (C) is the correct answer
A system is described by the inhomogeneous partial differential equation Sxx + 27,+/- = 41e-x/2sin( 3x)...
The twisting of a beam with rectangular cross-section is described by the inhomogeneous partial differential equation (PDE) below: 024 049 = -2 əx2 + ayż Eqn 2.1 where x and y are the coordinates of the cross-section and p(x,y) is the warp or distortion of the cross-section. The cross-section is bounded by –p sx sp and —q sy sq. The boundary conditions are given by: 0(p,y) = 0, 4(-p,y) = 0, 4(x,q) = 0 and 4(x,-q) = 0. Using the...
find the general solution of the differential equation by using the system of linear equation. please need to be solve by differential equation expert. d^2x/dt^2+x+4dy/dt-4y=4e^t , dx/dt-x+dy/dt+9y=0 Its answer will look lile that: x(t)= c1 e^-2t (2sin(t)+cos(t))+ c2 e^-2t (4e^t-3sin(t)-4cos(t))+ 20 c3 e^-2t(e^t-sin(t)-cos(t))+2 e^t, y(t)= c1 e^-2t sin(t)+ c2 e^-2t(e^t-2sin(t)-cos(t))+ c3 e^-2t(5e^t-12sin(t)-4cos(t))
Consider the partial differential equation, with the initial condition: 1 2yuz + 3x?uy = 9x?y?, u(x,0) = x3 + 1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.
PROBLEMS Solve for y. 3.1. - x + 4x + sin 6x 3.4. y + 3x = 0 3.5. (x-1)? ydx + x? (y - 1)dy = 0 Just find a solution. Solving for y is tough. Test for exactness and solve if exact. 3.6. (y - x) dx + (x? - y) dy - 0 3.7. (2x + 3y) dx + (3x + y - 1) dy - 0 3.8. (2xy Y + 2xy + y) dx + (x*y*el...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck.
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...
c). X*+16% +63=0 ). Please Solve for . For some problems, tell which results are the only Solutions; or if there is no solution. a). .01.2¥ r.8)=-4058 - b) flors)- $ (3x +27) d). £472-94 +36=0 e). VX +21 = 8 f). √3 x 9 = -6 8). +6=7-6 3). , * I). (**-+|-} k). 16"*998- 1). 7"-1 m. lax + ln (x-2) = ln (x+4) ►) 4500's - 3 = (Fid Pract values, for 2x 360) 0). V cscx...
Exercise 3 Given is the following partial differential equation: Show that w(x, y, z)= sin(52) is a solution of this partial differential equation. Exercise 4 Given is a three-dimensional volume enclosed by the planes y=0,2 = 0, y = and z=a-x+y, with a > 0 a constant. -x (4a) Make a three-dimensional sketch of this volume. Clearly indicate all characteristic features. (4b) Give an integral, with integration boundaries, that can be used for calculating the volume of the object. (4c)...
7. Show that the equation f(x) = x^3 + 3x^2 - 9x + 7 = 0 has a solution for some x is E(-6; -5). Apply Newton’s method with an initial guess x0 = -5 to find x2. 8. Find the intervals of increase and decrease of the function x2e^-2x. 9. Sketch the graph of the curve y = x3 + 3x2 - 9x + 7. Be sure to find the intervals of increase, decrease and constant concavity and all...
Show ALL Work c), x+16% +63 = 0 D. Please Solve for x. For some problems, tell which results are the only solutions; or if there is no solution. a). .010.2x +.8)=.005% -1 b). {1x+5) = $ (3x +27) d) x ² 48²_9x+36=0 e). √x+21=8 f). Ý3x-9 =-6 g). √x+6 = x-6 h), biz * * -- I), X J. 1 3 3x - 1 + 1 = 2 / 4 K). 16 = 8 2x-1 2). 2 9 +...