Chapter 6: Q14E (page 332)

Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 2 y''' + 10 y'' + 18 y' + 9 y = 0$

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variable is $y (x) = c_{1} e^{- x} + c_{2} x e^{- x} + c_{3} \cos 3 x + c_{4} \sin 3 x$

Step by step solution

Auxiliary equation:

Consider the equation . $y^{(4)} + 2 y''' + 10 y'' + 18 y' + 9 y = 0$

The associated auxiliary equation is $r^{4} + 2 r^{3} + 10 r^{2} + 18 r + 9 = {(r + 1)}^{2} (r^{2} + 9) = 0$ which has $r = - 1$ as a double solution and $r = - 3 i$ and $r = 3 i$ as a simple solutions.

General solution:

The general solution to the given equation is given by

$y (x) = c_{1} e^{- x} + c_{2} x e^{- x} + c_{3} \cos 3 x + c_{4} \sin 3 x$

Hence the final solution is $y (x) = c_{1} e^{- x} + c_{2} x e^{- x} + c_{3} \cos 3 x + c_{4} \sin 3 x$

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Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 2 y''' + 10 y'' + 18 y' + 9 y = 0$

Short Answer

Step by step solution

Auxiliary equation:

General solution:

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Find a general solution for the differential equation with x as the independent variable:y(4)+2y'''+10y''+18y'+9y=0

Short Answer

Step by step solution

Auxiliary equation:

General solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Calculus

Decision Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 2 y''' + 10 y'' + 18 y' + 9 y = 0$