Chapter 8: Q5.9P (page 414)

Solve the following differential equations by the methods discussed above and compare computer solutions. $(D^{2} - 4 D + 13) y = 0$

Short Answer

Expert verified

The solution is $y = e^{2 x} (c_{1} e^{3 i x} + c_{2} e^{- 3 i x})$

Step by step solution

Given information

A differential equation is given as $(D^{2} - 4 D + 13) y = 0$ .

Differential equation.

A differential equation of the form $(D - a) (D - b) y = 0, a, b = α \pm β i$ has general solution

$y = c_{1} e^{(α + β i) x} + c_{2} e^{(α - β i) x}$

Find the solution of the given differential equation.

Suppose that $D = \frac{d}{d x}$

Then

$D y = \frac{d y}{d x} D y = y^{'} D^{2} y = \frac{d}{d x} (\frac{d y}{d x}) \frac{d^{2} y}{d x^{2}} = y^{''}$

Given differential equation is $(D^{2} - 4 D + 13) y = 0$ .

Auxiliary equation is

$D^{2} - 4 D + 13 = 0 D = \frac{- (- 4) \pm \sqrt{{(- 4)}^{2} - 4 \cdot 1 \cdot 13}}{2 \cdot 1} D = \frac{4 \pm 6 i}{2} D = 2 + 3 i D = 2 - 3 i$

These are unequal roots.

$\Rightarrow D = 2 + 3 i$ and $D = 2 - 3 i$

Differential equation becomes -

$\Rightarrow [D - (2 + 3 i)] [D + (2 - 3 i)] y = 0$

These are separable equation with solution

$y = c_{1} e^{(2 + 3 i) x} and y = c_{2} e^{(2 - 3 i) x}$

The general solution will be given by

$y = c_{1} e^{(2 + 3 i) x} + c_{2} e^{(2 - 3 i) x} y = e^{2 x} (c_{1} e^{3 i x} + c_{2} e^{- 3 i x})$

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Solve the following differential equations by the methods discussed above and compare computer solutions. $(D^{2} - 4 D + 13) y = 0$

Short Answer

Step by step solution

Given information

Differential equation.

Find the solution of the given differential equation.

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Solve the following differential equations by the methods discussed above and compare computer solutions.D2-4D+13y=0

Short Answer

Step by step solution

Given information

Differential equation.

Find the solution of the given differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Physics of Motion

Magnetism

Geometrical and Physical Optics

Dynamics

Fields in Physics

Study anywhere. Anytime. Across all devices.

Solve the following differential equations by the methods discussed above and compare computer solutions. $(D^{2} - 4 D + 13) y = 0$