Write the function f(x) = x(1-x) as an eigenfunction expansion of the eigenfunctions that correspond to the Sturm-Liouville problem: y" + 2y = 0, y(o)=0, y'(1) =0

Find the general solution of the DE: y’’’ - 4y’ = 32xe^2x - 24x^2

Let y₁ = xe-* be a solution of the equation y" + 2y' + y = 0. Find the second solution from y₁ by reduction of order (apply formula), and then find the general solution of the equation.

Suppose 2 = f(). (a) Given y(xo) = = Yo, when does this equation have a unique solution? (b) Make a substitution z = how to separate the variables. z(x, y) to transform this equation into a separable one. Show (c) Construct an example of such an equation and solve it using the substitution found in (b).

2 )) Solve: x'y" - 7xy' + 41y =0

Find the general solution y''-y'-2y=0

Write down two linearly independent solutions of the Bessel equation 16x?y" + 16ry + (162 – 9) y = 0.

Y₁ = 1+ a3x³ + A6㺠+... Y₂ = x + b₁x¹ +b7x7 +... Enter the first few coefficients: Az = a6 = b₁ b7 ) Find two linearly independent solutions of y" + 6xy = = - 0 of the form

18. Determine the roots of the auxiliary equation of y" +3y" – 4y' – 12y = 0 - (a) -3, –2, 2 (b) -3, –2, 3 (с) 0, 2, —2 (d) -3,2,3

Eliminate the arbitrary constant on the given equation: y = C1r + C2e O (2 - 22) y" + (2 + 2) y - (2« – 2) y O (3r - 42) y" - (a + 4) y (4x - 4) y 0 lin O (2? + 2x) y" + (z2 - 2) y-(2x + 2) y = 0 MIRI O (z – 22) / + ( – 4) / 2ar) 3f + ( 4) -(2)y-0