Chapter 4: Q4.3-4E (page 172)

The auxiliary equation for the given differential equation has complex roots. Find a general solution. $y'' - 10 y' + 26 y = 0$

Short Answer

Expert verified

The auxiliary equation for the given differential equation $y'' - 10 y' + 26 y = 0$ has complex roots and its general solution is $y (t) = e^{5 t} (c_{1} \cos (t) + c_{2} \sin (t))$ .

Step by step solution

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $α \pm iβ$ , then the general solution is given as:

$y (t) = c_{1} e^{αt} cosβt + c_{2} e^{αt} sinβt$

The auxiliary equation.

Assume that $y = e^{rt}$ is a solution to the given equation.

Since the given equation is of order two, differentiate role="math" localid="1654061985491" $y$ with respect to role="math" localid="1654061981089" $x$ twice:

$y' (t) = r e^{rt} y^{"} (t) = r^{2} e^{rt}$

Substitute $y ", y'$ and $y$ in the given equation to obtain: $e^{rt} (r^{2} - 10 r + 26) = 0 .$

Then the auxiliary equation is $r^{2} - 10 r + 26 = 0 .$

Finding the roots.

Solve for the roots of the auxiliary equation

$r_{1 / 2} = \frac{10 \pm \sqrt{10^{2} - 4 \times 1 \times 26}}{2 \times 1} = \frac{10 \pm \sqrt{100 - 104}}{} = \frac{10 \pm 2 i}{} = 5 \pm i$

Final answer.

Since it has a complex conjugate of the form $r = α \pm iβ$ for $α = 5$ and $β = 1$

Thus, the general solution is $y (t) = e^{5 t} (c_{1} \cos (t) + c_{2} \sin (t))$ .

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The auxiliary equation for the given differential equation has complex roots. Find a general solution. $y'' - 10 y' + 26 y = 0$

Short Answer

Step by step solution

Complex conjugate roots.

The auxiliary equation.

Finding the roots.

Final answer.

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The auxiliary equation for the given differential equation has complex roots. Find a general solution.y''-10y'+26y=0

Short Answer

Step by step solution

Complex conjugate roots.

The auxiliary equation.

Finding the roots.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Calculus

Discrete Mathematics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

The auxiliary equation for the given differential equation has complex roots. Find a general solution. $y'' - 10 y' + 26 y = 0$