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Find a homogeneous linear differential equation with constant coefficients whose general solution is given:

y = C₁ cos(x) + C₂ sin(x) + C₃ cos(8x) + C₄ sin(8x)

A. y" + 9y" + 8y' = 0
B. y" - 9y' + 8y = 0
C. y⁽⁴⁾ - 65" + 64y = 0
D. y⁽⁴⁾ + 65y" + 64y = 0
E. y" + 65y' + 64y = 0

Sagot :

masifakrami

Answer:

The correct answer is:

C. [tex]\( y^{(4)} + 65y'' + 64y = 0 \)[/tex]

Step-by-step explanation:

To find a homogeneous linear differential equation with constant coefficients whose general solution is [tex]\( y = C_1 \cos(x) + C_2 \sin(x) + C_3 \cos(8x) + C_4 \sin(8x) \),[/tex] we must determine the characteristic equation corresponding to this solution.

Given the solution components:

[tex]1. \( \cos(x) \) and \( \sin(x) \)\\2. \( \cos(8x) \) and \( \sin(8x) \)[/tex]

The characteristic roots corresponding to these solutions are:

[tex]1. \( \cos(x) \) and \( \sin(x) \) imply roots \( \pm i \)\\2. \( \cos(8x) \) and \( \sin(8x) \) imply roots \( \pm 8i \)[/tex]

Thus, the characteristic polynomial must have roots [tex]\( \pm I \) and \( \pm 8i \). This gives us the characteristic equation:\[ (r - i)(r + i)(r - 8i)(r + 8i) = 0 \][/tex]

The characteristic polynomial corresponding to these roots is:

[tex]\[ (r^2 + 1)(r^2 + 64) = 0 \][/tex]

Expanding this product, we get:

[tex]\[ (r^2 + 1)(r^2 + 64) = r^4 + 64r^2 + r^2 + 64 = r^4 + 65r^2 + 64 \][/tex]

Therefore, the corresponding differential equation is:

[tex]\[ y^{(4)} + 65y'' + 64y = 0 \][/tex]

So, the correct answer is:

C. [tex]\( y^{(4)} + 65y'' + 64y = 0 \)[/tex]

SophiaJudy

Homogeneous linear differential equation with constant coefficients D. y⁽⁴⁾ + 65y'' + 64y = 0.

To find a homogeneous linear differential equation with constant coefficients whose general solution is given by:

y = C₁ cos(x) + C₂ sin(x) + C₃ cos(8x) + C₄ sin(8x),

  1. we need to identify the characteristic equation corresponding to this general solution.
  2. The characteristic equation arises from the roots of the homogeneous differential equation.
  3. The given solution comprises terms involving cos(x), sin(x), cos(8x), and sin(8x).
  4. This implies that the characteristic roots are ±i and ±8i, where i is the imaginary unit.

The corresponding characteristic equation will then be:

  • (r - i)(r + i)(r - 8i)(r + 8i) = 0

Expanding this, we get:

  • (r² + 1)(r² + 64) = 0

This can be rewritten as:

  • r⁴ + 65r² + 64 = 0

The corresponding differential equation with these characteristic roots is:

  • y⁽⁴⁾ + 65y" + 64y = 0.
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