Use separation of variables to find a solution of the partial differential equation ди ди x² მე + y² = მყ = 0, on x, y € (0,∞), with boundary value u(x, 1) = e(1-x)/x¸

By using separation of variables u(x, t) = X (x)T(t), break the partial differ- ential equation ди + u = 4 into two ordinary differential equations. Your answer may contain some constant ).

Use separation of variables to find any solution to the partial differential equation uxt=x²e4t

Consider the differential equation: (56x^2)y″−16x(x+7)y′+16(x+7)y=5x3, x>0.   You can verify that y1=2x and y2=5xexp⁡(2x/7) satisfy the corresponding homogeneous equation.   Compute the Wronskian W between y1 and y2 Apply variation of parameters to find a particular solution.

Using the substitution y=vxanddy=vdx+xdv, solve the homogeneous differential equation [xsin(-)-y]dx + |dx + xdy=0 y y ℗ x (exe ² - co₁²-) = c A CSC X X ℗ B y y x (exc²-=-=- co₁²-) = y csc X X y © > [exe ² + C y X =C y + cor² =) = c X y y Ⓒ + x[cx ² + + CO ² ) = C D X CSC X Xx

Solve

Obtain the differential equation of all straight lines at a fixed distance p from the origin * O (xy-y)*2=p[1-(v)*2] O (xy'+y)*2=[1+()^21 O (xy-y)*2=p[1+(y)*2] O cxy'+y)^2=[1+yl

Solve the general solution of the second order non-homogeneous linear differential equation using the "transformation of parameters" metho

If the function f(x,y) is continuous near the point (a,b), then at least one solution of the differential equation y' = f(x,y) exists on some open interval I containing the point x = a af and, moreover, that if in addition the partial derivative is continuous near (a,b) then this solution is unique on some (perhaps smaller) interval J. Determine whether ду existence of at least one solution of the given initial value problem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed dy 3, dx = √yy(0) = 4 .. O A. The theorem implies the existence of at least one solution because f(x,y) is continuous near the point continuous near that same point. OB. The theorem implies the existence of at least one solution because f(x,y) is continuous near the point of because -= is not continuous near that same point. dy C. The theorem does not imply the existence of at least one solution because f(x,y) is not continuous near the point (0,4) This solution is unique because Jf…

If the function f(x,y) is continuous near the point (a,b), then at least one solution of the differential equation y' = f(x,y) exists on some open interval I containing the point x = a and, moreover, that if in of addition the partial derivative is continuous near (a,b) then this solution is unique on some (perhaps smaller) interval J. Determine whether existence of at least one solution of the given initial ду value problem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. dy dx = B. 6x²y^; y(7) = -2 Select the correct choice below and fill in the answer box(es) to complete your choice. (Type an ordered pair.) A. The theorem implies the existence of at least one solution because f(x,y) is continuous near the point This solution is unique because af ду The theorem implies the existence of at least one solution because f(x,y) is continuous near the point continuous near that same point. C. The theorem does not imply the existence of at least one…