Use (20) in Section 6.4. 1 - 2a y" + + (b²c²x²c − 2 + 0² - p²c²)y ₁ X Find the general solution of the given differential equation on (0, ). (The definitions of various Bessel functions are given here.) xy" + 2y' + 9y = 0 ○ y(x) = x¹/² [C₁J₁ (3x −¹/2) + C₂Y₁(3x-¹/2)] Ⓒy(x) = x-¹/²[C₁³₁ (3x¹/2) + C₂Y₁(3x¹/2)] O y(x) = x-1/² [C₁J₁ (6x¹/2) + C₂Y₁(6x¹/2)] O y(x) = x-¹/² [C₁J₁ (3x¹/2) + C₂J_1(3x¹/2)] 1/2. -1/2. y = 0, p≥0 (20) -1/2..

The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, e-JP(x) dx Y2 = Y1(x) (5) as instructed, to find a second solution y2(x). x²y" – 11xy' + 36y = 0; Y1 =x6 Y2 = In(\x\)

Use (1) in Section 6.4. xzy" + xy' + (x2 – v²)y = 0 (1) Find the general solution of the given differential equation on (0, ). (The definitions of various Bessel functions are given here.) x²y" + xy' + (x -36 36/ O yX) = C,J1/6(x) + C_1/6(x) O XX) = C,J1/6(x) + C,)_1/6(-x) o xX) = CJ1/36(x) + C,)_1/36(-x) O y(x) = C,11/6(x) + C,K1/6(x) O YX) = C,J1/36(x) + C_1/36(x) = 0 %3D %3D %3D %3D

The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, e-SP(x) dx Y2 = Y1(x) (5) as instructed, to find a second solution y2(x). xy" + y' = 0; Yı = In x Y2 =