Chapter 4: Q41E (page 165)

Find three linearly independent solutions (see Problem35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.
y"' + 3y" - 4y' - 12y = 0.

Short Answer

Expert verified

The three independent solutions of the given differential equation are:

y₁(t) = e^2t,y₂(t) = e^-2t and y₃(t) = e^-3t

The general solution is y(t) = c₁e^2t + c₂e^-2t + c₃e^-3t

Step by step solution

Finding the roots of the auxiliary equation.

The given differential equation is y"' + 3y" - 4y' - 12y = 0

The auxiliary equation for the given differential equation is r³ + 3r² - 4r - 12 = 0

Finding the roots of the auxiliary equation.

$\begin{array}{rcl} r^{3} + 3 r^{2} - 4 r - 12 & = & 0 \\ r^{2} (r + 3) - 4 (r + 3) & = & 0 \\ (r - 2) (r + 2) (r + 3) & = & 0 \\ r & = & 2, - 2, - 3 \end{array}$

Hence, the roots for the auxiliary equation are 2,-2 and -3.

Substitute the root values

Therefore the three independent solutions of the given differential equation are:

y₁(t) = e^2t,y₂(t) = e^-2t and y₃(t) = e^-3t

Therefore, the general solution is y(t) = c₁e^2t + c₂e^-2t + c₃e^-3t.

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Find three linearly independent solutions (see Problem35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.
y"' + 3y" - 4y' - 12y = 0.

Short Answer

Step by step solution

Finding the roots of the auxiliary equation.

Substitute the root values

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Find three linearly independent solutions (see Problem35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.y"' + 3y" - 4y' - 12y = 0.

Short Answer

Step by step solution

Finding the roots of the auxiliary equation.

Substitute the root values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Statistics

Discrete Mathematics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Find three linearly independent solutions (see Problem35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.
y"' + 3y" - 4y' - 12y = 0.