12. Solve the problem: Utt = c*upx + Ax, 0 < x < 1, t > 0, A= constant, и (х,0) — 0, и (0, t) — 0, Ut (x, 0) = 0, 0< x < 1, u (1, t) = 0, t > 0.

Q: Solve the initial value problem y + y = g(t), where 1, g(t) ={ 0, The solution should be continuous…

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Q: Show that the general solution of y(4) − y = 0 can be written as y=c1cost+c2sint+c3cosht+c4sinht.…

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Q: Suppose that the function y(x) = E Cnx" solves the initial value problem n=0 y" (x) – xy'(x) – y(x)…

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Q: 2. Let y(x) = tan r. Show that y is the solution to the initial-value problem %3D dy =1+ y, y(0) =…

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Q: 4. Solve the equation satisfying the conditions: dt ax (i) T=0 when x= 0 and 1 for all t 2х, 0Sx<1/2…

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Q: B/Verify that u(x, y, z) = Tr + yk 4+z? satisfies the partial differential equation a2u a2u a2u = 0…

A: B.

Q: Solve the Neumann problem Ди — 0 Uz(0, y) = 0, Uz(7, y) = D, cos(ny) Uy(x, 0) = 0, u,(x, T) = 0 0 <…

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Q: 5. Solve the initial value problem y' = t*t, y(0) = 1, where * is the convolution operator.

A: By taking Laplace transformation and inverse Laplace transformation we can solve this question

Q: 2. Verify that y = -t cos t – t is a solution of the initial-value problem dy = y + t²sin t dt y(7)…

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Q: Use Algorithm 12.1 to approximate the solution to the elliptic partial differential equa a?u a?u + =…

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Q: Proceed as in Example 6 in Section 2.3 to solve the given initial-value problem. S 1, 0sx s 1 x > 1…

A: Given that dydx+y=fx  , y0=1  fx=1  ,0≤x≤1-1    x&gt;1Solution:When 0≤x≤1dydx+y=1 this is the linear…

Q: 2. Find the particular solution using the initial condition. ry'-y = x,y(1) = 7.

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Q: S Ive the Neumann problem { 0 < x, Y < 0 < y < T 0 < x < T || cos(ny) Uz(0, y) = 0, Uz(7, y) = E1…

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Q: 16 Q4- Solve the following system of ODE dx dt Eaux"> dy dt = y + 1 =x+1

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Q: Solve the initial-value problem with the initial condition Yo = 0. dy y + 3 dt %3D y =

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Q: Solve the fallowing Cauchy problem U4- Au + cu=0 in R" x(0, 0) On IR"x{t=0}. U = where c is a…

A: Given the Cauchy problem as ut-∆u+cu=0,             in ℝn×0,∞                (1) re-write the…

Q: 2 4 find the orthogonal trajectories to c¯y y" = 4y´ + x´ 2 2 .... ..... ..... .....

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Q: () 2. If k = w3 – wv², w=xy, v = w+ x, find and using: dw a) the chain rule b) differentials.

A: Given: k=w3-wv2, w=xy, v=w+x   To find: (∂k∂w)x . (∂k∂x)w  using chain rule and differentials

Q: where the ends of the string are fixed, solve the initial-boundary value problems by the method of…

A: The given equation is a one-dimensional wave equation.  When both the ends of the string of length π…

Q: 1 So 2t en + e-2t ezt - e-at dt

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Q: 2nd Case (IBVP Summary) (Ut = 1.5uxx, 0 0 Ux(0, t) = 0, ux(6 ft, t) = 0, t > 0 (u(x,0) = 80°F, 0<…

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Q: 10. Solve the initial value problem dy dy +8 -9y 0 y(1) = 1, y'(1) = 0. %3D dt2 dt

A: We have given initial value problem, d2ydt2 + 8dydt - 9y =0 It's Auxiliary equation is given by, m2…

Q: Solve the separable initial value problem. 1. y' = In(x)(1+y²), y(1) = 1 → y= 2. y =…

A: Use the following formulas 1.∫xndx=xn+1n+1+c2.∫11+x2dx=tan-1x+c3,∫ln x dx=xln x-x+c

Q: Let y(t) be the solution of (x3 + 1) dy dt = y satisfying y(0) = 1. Compute approximations to…

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Q: 2. What is the solution to the initial-value problem PD.E. ut = c²uxx (u(x,0) = 0 lu,(x,0) = xe-x -…

A: Given that  utt=c2uxx    -∞&lt;x&lt;∞,  0&lt;t&lt;∞ux,0=0  and  utx,0=xe-x2,   -∞≤x&lt;∞ It is a…

Q: 5. Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium…

A: Given the heat equation is k∂2u∂x2-hu=∂u∂t, 0&lt;x&lt;L, t&gt;0 The initial temperature is f(x)…

Q: 6. Find the general solution to the DE. Show all of your work. dy =-0.2y+10 dt

A: Particular Differential Equation: A differential equation which is either in the form of , then the…

Q: 1/2 Q2. a) Using Hilbert-Schmidt method, solve y(x) = cos 2x + 2 f,"´ K (x, t)y(t)dt, where Ssin x…

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Q: (t -x)dt – tdx =0; when t = 1, x = 5

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Q: 2. Consider y+2-c y+1+cy, where e is a positive constant. For what values of e will the solution to…

A: Given yt+2-cyt+1+cyt

Q: Verify that u(r, t) = f(e-"x) solves the initial value problem (au + z8,u = 0, u(x,0) = f(z).

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Q: 8. Find the particular solution that satisfies the initial condition. y'+y = 6e*, Initial Condition:…

A: Find the particular solution

Q: 2. Find the first three approximations yı, y2, y3 using Euler's method for the initial value problem…

A: Let's find.

Q: 22) Use Euler method to estimate the solution of the ODE y' (xy?) ,y(1) = 2, h = 0.1 at x = 1.2

A: using the Euler's rule: yi+1=yi+hf(xi,yi) we will find the solution of given ODE we have…

Q: 2. Solve Un = Uxx X > 0, t> 0 with initial and boundary conditions given as u( x,0) = 1– exp(-x),…

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Q: Verify that u(z, t) = f(e'r) solves the initial value problem (a,u + z8,u = 0, u(x,0) = f(x).

A: Take the partial derivatives of u(x,t).

Q: 1. Use the Chain Rule to find 1. y = u² + 1 dy dx (show your complete solution) . u = x² - 1

A: dydx=4x3-4x

Q: 9. Find the solution to satisfying y = dy dt = 7y || y(6)=9

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Q: #3. Solve the fallowing Caucly problem in R" x(0,0) U4-Ou t cu = 0 u =9 On IR"x{t =0}. where c is a…

A: Given the Cauchy problem as  ut-∆u+cu=0,       in ℝn×0,∞           (1) re-write the equation (1) in…

Q: Minimize F(y, u) = f¿u²(t) dt under ý = -2y + u with y(0) = 1, y(1) = 0. Is your solution unique?

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Q: 7. Find the general solution to y" + 2(x – 1)y' + 4y = 0 using an integrating factor of the form ax…

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Q: 6. Solve the boundary-value problem u(x,0) =cos x, 0<x<1 u (0,t)=0, u, = Uxx + u; u(1,1)=0.

A: The equation is ut=uxx+u. Consider the solution ux, t=XxTt with X0=X1=0, T0=cosx. Then, we have…

Q: Let u (x, t) be the solution of the initial-boundary value problem Ut 0 0, Uz (0, t) = uz (4, t) =…

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Q: 7.Obtain the particular solution of (1 – xy)-2dx + [y? + x²(1 – xy)-²]dy = 0 when x = 2, y = 1

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Q: (j) Suppose that the solution of the initial value problem u" + u = 0.5 cost, u(0) = 0, u'(0) = 0 is…

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Q: 8. Find the solution y(t) of th d³y dy - 2y% = dt dt3

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Q: #3. Solve the fallowing Caucly problem in R" x(0, 0) Uf-Ou t cu =0 on IR"x{t =0}. U = where c is a…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.