5. Use the separation of variables method to determine the solution u(x; t) of the heat transfer equation du azu 0< x< 2, t> 0 at u(0; t) = u(2; t) = 0 ax2 %3D u(x; 0) = 2 sin n21 %3D

Q: etermine whether the function u(x, t)=ecos() is a solution of the heat conduction equa- on u₁=a²uxx-

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Q: 3. Find the total differential of z = f(x, y) if z2 - y = (1– 2x)z.

A: Total differential of a function z=fx,y is dz=∂z∂xdx+∂z∂ydy

Q: Determine the order of the DE. a.) y' + y = 5y^2 b.) (dy)/(dt) = t c.) y' = y'' + 5t^2

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Q: 4) Solve: x.[dy/dx] - y = x²sinx (Hint: Use the integrating factor).

A: Step 1: Step 2: Step 3: Step 4:

Q: w =sec (xyz) - Yx +y+z

A:

Q: Exercise (1): Determine the order and degree of the following differential equa- tions: (1) y - 4x +…

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Q: Determine the required integrating factor: y'=x-2y cot (2x) O sec (2x) O2cos (2x) O2cot (2x) O sin…

A: y' =x-2ycot2x

Q: Find the total differential. 7x + yl 10z - 5y W =

A: *Note – when finding the partial derivative w.r.t one variable, other variables consider as…

Q: Suppose F(x, y) = −y sin(x)i + cos(x)}. (a) Find a vector parametric equation for the parabola y =…

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Q: Solve the heat transfer equation for a metal rod in one dimension, assuming a constant cross-section…

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Q: The total differential of the function: z=x*e3y . .A dz = 3x "e3Y dx +4x³e3Ydy. .B dz= 3x"dx +e3ydy.…

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Q: 2. find the derivative of the function g(x, y, z) = 3e* cos yz,at the point in the direction of A =…

A:

Q: Find the image of y = 1 under f(z) =1/z %3D %3D

A: Image of y=1 under f(z)=1/z

Q: Graph the slope field for the equation dx/dy = -x(x2-4) and attach it to the solution. Draw (by hand…

A:

Q: Find all the second order partial derivatives of f(x.y)=ry +x*y' +x'y and %3D o f verify whether %3D…

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Q: Calculate dwldu and dwldv using the following functions: f(x, y. 2) = 3x² – 2ry+ 422 x = x(u, v) =…

A: