Problem #1 Y1(x)= x and Y2(x)=e* are linearly independent solution of the homogeneous equation: (x-1)y"-xy'+y =...
The indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, 0). Find the general solution of the given nonhomogeneous equation. *?y" + xy' + (x2 - 1)y = x3/2; Y1 = x-1/2 cos(x), Y2 = x-1/2 sin(x) y(x) =
A third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular solution satisfying the given initial conditions. yl) + 2y'' – y' - 2y = 0; y(0) = 2, y'(0) = 12, y''(0) = 0; Y1 = ex, y2 = e -X, y3 = e - 2x The particular solution is y(x) = .
A third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular solution satisfying the given initial conditions. y (3) + 2y" - y' - 2y = 0; y(0) = 7, y' (0) = 16, y''O) = 0; e y2 = e-X, y3 = e - 2x Y The particular solution is y(x) = .
Please prove this solution and explain why y2 can be taken as (x^2)(y1) Problem 2. Find the general solution of the equation Note that one of two linearly independent solutions is yi(r) -e. Solution. Using Abel's formula, we get the following relations for the Wronskian dW pi dW 2r1 On the other hand, Comparing these two expression for W(x), we can take y2 :- r2yı. Correspondingly, the general solution is Problem 2. Find the general solution of the equation Note...
4. Verify that yi xPJp(x) and y2 - xPYp(x) are linearly independent solutions of xy" + (1-2 )y, + xy-0, x > 0. 4. Verify that yi xPJp(x) and y2 - xPYp(x) are linearly independent solutions of xy" + (1-2 )y, + xy-0, x > 0.
lPLS SHOW ALL WORK Problem 6. Prove that the family of functions {Y1 = 1, y2 = e", y3 = 222} is linearly independent on (-00,00). Find a homogeneous linear equation whose general solution is y = C1 + C2e? + C2e2^ .
17. Another way to check if y1, y2 are linearly INDEPENDENT in an interval I is: for all I for all I does not exist for all I d. none of the above 18. If y1 is a solution of the equation y "+ P (x) y '+ Q (x) y = 0, a second solution would be y2 (x) = u (x) y1 (x) where u (x) it is: d. all of the above 19. The following set...
Consider the homogeneous linear third order equation A) xy'''−xy'' + y'−y = 0 Given that y1(x) = e^x is a solution. Use the substitution y = u*y1 to reduce this third order equation to a homogeneous linear second order equation in the variable w = u'. You do not need to solve this second order equation. B.) xy''' + (1−x)y'' + xy'−y = 0. Given that y1(x) = x is a solution. Use the substitution y =...
Verify that the given functions Y1 and y2 satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneous equation. x2y" – 3xy' + 4y = 7x? In x, x>0; 71(x) = x2, yz(x) = x2 In x Y(x) =
Part A is first 2 lines, Part B is last 2 lines, thanks! For Problems 13-17, find a particular solution of the nonhomogeneous equation, given that the functions y(x) and y2(x) are linearly independent solutions ofthe corresponding homogeneous equation. Note: The cocfficient of y" must always be 1, and hence a preliminary division may be required y2(x) = x-2 ·y1(x) = x y2(x) = ex For Problems 13-17, find a particular solution of the nonhomogeneous equation, given that the functions...