2 : Let า1(r,y) bc a harmonic function and Lct F (u:uy, llzu,) bc a vector field. Show that if uzy is positive everywhere and u(z, y) has no local extrema then F is a source.
2 : Let า1(r,y) bc a harmonic function and Lct F (u:uy, llzu,) bc a vector field. Show that if uzy is positive everywhere and u(z, y) has no local extrema then F is a source.
Solve the following partial differential equations using the
Laplace transform method.
x〉o, 2 5 ,>0 lim u(x, ,) = 0, u(0,,)-1, 3) İhu Ot
x〉o, 2 5 ,>0 lim u(x, ,) = 0, u(0,,)-1, 3) İhu Ot
ANSWER BOTH PARTS
» Suppose that u depends on z and w, while z and w depend on x and y, where both x and y depend on t. Draw a tree diagram and write the chain rule for computing อน ot ·Explain why the direction of maximal ascent coincides with the direction of the gradient
» Suppose that u depends on z and w, while z and w depend on x and y, where both x and y depend...
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) =...
Select the Boolean expression that is not satisfiable. 3 (z+u)(z+x)(z+x")(u+y)(u+y) (z+u")(z+x)(z+x)(u+y)(u+y) (zº+u)(z+x)(z+x")(u+y)(u+y") (z+u")(z+x)(z+x")(u+y)(u+y") 6 a Question 13 (1 point) Select the statement that is not a proposition. 12 5+4 = 8 It will be sunny tomorrow. 15 Take out the trash. 7 18 Chocolate is the best flavor. 20 21 Question 14 (1 point) p = T. q = F, and r = F. Select the expression that evaluates to true. 23 24 Срла -р avr
Assuming f E C3(R3) and g E C23) in C2 (R3 x (0, oo). , show that u E u(x, t)- ot 4Tc2t X = (2.1, 22, 23) e R3
Assuming f E C3(R3) and g E C23) in C2 (R3 x (0, oo). , show that u E u(x, t)- ot 4Tc2t X = (2.1, 22, 23) e R3
*Note: Please answer all parts, and explain all workings. Thank
you!
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
IVI U OT 6.00 p Flag question If a and b are constants, the solution of (aye axy+b)dx+(2ye xy + axy’e x – 1)dy=0 is Select one: a.yle axy + bx-y=c b. eaxy + bx= c O c. ye axy + bx - y = c d. y2e xy + x - y=c e. ye ax + bx-y=C f. ye xy +bx+y=c g. y? (e axy - 1) = 0 h. Y x + bx = y
A) Suppose U=X∙Y3. Find X* and Y*. B) Suppose U=X3∙Y. Find X* and Y*. C) Suppose U=X3∙Y2. Find X* and Y*. D) Suppose U=X∙Y5. Find X* and Y*.
Let u = u(x,y) and x = x(r,9), y = y(r,). ди ди a. Let x = r cos Q, y = r sin p. Find and a2u ar' 29 ar2 b. u = -x x = r sin 29,y = r tan’ 4, P (1,5). Find ou at the point P. де до