Differential Equations - Mathematics Assignment Help

Assignment Task

Differential Equations. 

(1) Make a substitution* to linearise the Ricatti equation 2f2 = Ax, where A is a constant. Hence solve the equation. 

(2) Find the general solution of the equation x2y" — 3xy' + 3y = 2x4 cos x. You may use the fact that y(x) = x is a solution of the homo-geneous problem for extra credit. 

(3) Solve the equation (1 — x2)y" — 2xy' + 20y = 0 by means of a power series expansion. 

(4) Use the method of Frobenius to solve the equation x2y" + 2xy' + (x + 3)y = O. The Gamma function will help here. Express the solution in terms of Bessel functions. 

(5) Let yi , y„ be linearly independent solutions of the linear ODE an(x)y(") + • • + ao(x)y = 0, x E / C R. Show that yi, y„ cannot be simultaneously equal to zero. That is, there does not exist a E I such that yi (a) = • • • y„ (a) = O. 

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