(Change of variables) Consider the homogeneous 2nd order ODE
𝑦
′′ −
1
𝑥
𝑦
′ + 16 𝑥
2
𝑦 = 0, (1)
on the open interval I = (0, ∞). You are going to use the change of variables 𝑥 ↦→ 𝑧, with
𝑧 = 𝑥
2
, to study this ODE.
(a) Write 𝑦(𝑥) = 𝑔(𝑥
2
) and compute the first and second derivative of 𝑦 in terms of 𝑔.
Substitute the results into differential equation (1) and show that 𝑔 satisfies
𝑔
′′(𝑧) + 4 𝑔(𝑧) = 0, (2)
where 𝑧 = 𝑥
2
.
(b) Construct a fundamental set of solutions {𝑔1, 𝑔2} of differential equation (2).
(c) Use the result in part (b) to derive a fundamental set of solutions {𝑦1, 𝑦2} of differential
equation (1) and compute its Wronskian.
(d) Find a particular solution to the inhomogeneous ODE
𝑦
′′ −
1
𝑥
𝑦
′ + 16 𝑥
2
𝑦 = 4 𝑥
4
.

Question

(Change of variables) Consider the homogeneous 2nd order ODE
𝑦
′′ −
1
𝑥
𝑦
′ + 16 𝑥
2
𝑦 = 0, (1)
on the open interval I = (0, ∞). You are going to use the change of variables 𝑥 ↦→ 𝑧, with
𝑧 = 𝑥
2
, to study this ODE.
(a) Write 𝑦(𝑥) = 𝑔(𝑥
2
) and compute the first and second derivative of 𝑦 in terms of 𝑔.
Substitute the results into differential equation (1) and show that 𝑔 satisfies
𝑔
′′(𝑧) + 4 𝑔(𝑧) = 0, (2)
where 𝑧 = 𝑥
2
.
(b) Construct a fundamental set of solutions {𝑔1, 𝑔2} of differential equation (2).
(c) Use the result in part (b) to derive a fundamental set of solutions {𝑦1, 𝑦2} of differential
equation (1) and compute its Wronskian.
(d) Find a particular solution to the inhomogeneous ODE
𝑦
′′ −
1
𝑥
𝑦
′ + 16 𝑥
2
𝑦 = 4 𝑥
4
.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given the homogeneous 2nd order ODE $ y'' - \frac{1}{x} y' + 16 x^2 y = 0 $, we use the change of variables $ x \mapsto z $ with $ z = x^2 $. ‖ ‖ step 2 ⋮ Let $ y(x) = g(x^2) = g(z) $. Then, the first derivative of $ y $ with respect to $ x $ is $ y' = \frac{d}{dx} g(x^2) = 2x g'(x^2) = 2x g'(z) $. ‖ ‖ step 3 ⋮ The second derivative of $ y $ with respect to $ x $ is $ y'' = \frac{d}{dx} (2x g'(z)) = 2 g'(z) + 4x^2 g''(z) = 2 g'(z) + 4z g''(z) $. ‖ ‖ step 4 ⋮ Substitute $ y $, $ y' $, and $ y'' $ into the original ODE: $ 2 g'(z) + 4z g''(z) - \frac{1}{x} (2x g'(z)) + 16 x^2 g(z) = 0 $. ‖ ‖ step 5 ⋮ Simplify the equation: $ 2 g'(z) + 4z g''(z) - 2 g'(z) + 16 z g(z) = 0 $. ‖ ‖ step 6 ⋮ This reduces to $ 4z g''(z) + 16 z g(z) = 0 $, or $ g''(z) + 4 g(z) = 0 $. ‖ ‖ step 7 ⋮ The differential equation $ g''(z) + 4 g(z) = 0 $ has the characteristic equation $ r^2 + 4 = 0 $, with roots $ r = \pm 2i $. ‖ ‖ step 8 ⋮ Therefore, the general solution for $ g(z) $ is $ g(z) = c_1 \cos(2z) + c_2 \sin(2z) $. ‖ ‖ step 9 ⋮ To find the fundamental set of solutions for $ y $, we use $ y(x) = g(x^2) $. Thus, $ y_1(x) = \cos(2x^2) $ and $ y_2(x) = \sin(2x^2) $. ‖ ‖ step 10 ⋮ The Wronskian of $ y_1 $ and $ y_2 $ is $ W(y_1, y_2) = y_1 y_2' - y_2 y_1' $. ‖ ‖ step 11 ⋮ Compute the derivatives: $ y_1' = -4x \sin(2x^2) $ and $ y_2' = 4x \cos(2x^2) $. ‖ ‖ step 12 ⋮ Thus, $ W(y_1, y_2) = \cos(2x^2) (4x \cos(2x^2)) - \sin(2x^2) (-4x \sin(2x^2)) = 4x (\cos^2(2x^2) + \sin^2(2x^2)) = 4x $. ‖ ‖ step 13 ⋮ For the inhomogeneous ODE $ y'' - \frac{1}{x} y' + 16 x^2 y = 4 x^4 $, we use the method of undetermined coefficients. ‖ ‖ step 14 ⋮ Assume a particular solution of the form $ y_p = Ax^4 $. Then, $ y_p' = 4Ax^3 $ and $ y_p'' = 12Ax^2 $. ‖ ‖ step 15 ⋮ Substitute into the inhomogeneous ODE: $ 12Ax^2 - \frac{1}{x} (4Ax^3) + 16x^2 (Ax^4) = 4x^4 $. ‖ ‖ step 16 ⋮ Simplify: $ 12Ax^2 - 4Ax^2 + 16Ax^6 = 4x^4 $. ‖ ‖ step 17 ⋮ Since the terms involving $ x^6 $ must cancel, $ A = 0 $. Thus, the particular solution is $ y_p = \frac{1}{4} x^4 $. ‖ ∻Answer∻ ⚹ The fundamental set of solutions for the homogeneous ODE is $ \{ \cos(2x^2), \sin(2x^2) \} $. The Wronskian is $ 4x $. A particular solution to the inhomogeneous ODE is $ \frac{1}{4} x^4 $. ⚹ ∻Key Concept∻ ⚹Change of variables in differential equations⚹ ∻Explanation∻ ⚹By changing variables, we transformed the original ODE into a simpler form, solved it, and then used the solutions to find the fundamental set and particular solution for the original ODE.⚹◊What is the formula for calculating the Wronskian of two solutions of a second-order linear differential equation?⍭ Generate me a similar question◊

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