Chapter 6: Q5E (page 341)

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
$y^{'''} + y^{'} = t a n x$

Short Answer

Expert verified

The particular solution is $y_{p} = \ln | \sec x | - \sin x \ln | \sec x + \tan x |$

Step by step solution

Definition

Variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.

Find complementary solution

The given equation is: $y^{'''} + y^{'} = \tan x$

The auxiliary equation is $D^{3} + D = 0$

So, $D = \pm i, 0$

So ${1, \cos x, \sin x}$ fundamental set.

Calculate Wornkians

The value of wronkians is:

$W [\begin{matrix} 1 & \cos x & s i n x \end{matrix}] = |\begin{matrix} 1 & \cos x & \sin x \\ 0 & - \sin x & \cos x \\ 0 & - \cos x & - \sin x \end{matrix}| = 1$

$W_{1} = (- 1)^{3 - 1} W [\begin{matrix} \cos x & \sin x \end{matrix}] = |\begin{matrix} \cos x & \sin x \\ - \sin x & \cos x \end{matrix}| = 1 W_{2} = (- 1)^{3 - 2} W [\begin{matrix} 1 & \sin x \end{matrix}] = - \cos x W_{3} = (- 1)^{3 - 3} W [\begin{matrix} 1 & \cos x \end{matrix}] = - \sin x$

For particular solution

The particular solution is given by:

$y_{p} = 1 \int \frac{1 (\tan x)}{1} d x + \cos x \int \frac{- \cos x (\tan x)}{1} d x + \sin x \int \frac{- \sin x \cdot \tan x}{1} d x y_{p} = \ln | \sec x | + \cos^{2} x + \sin x I_{1} I_{1} = \int - \sin x \cdot \tan x d x = \sin x - \ln (\sec x + \tan x)$

$y_{p} = \ln | \sec x | + \cos^{2} x + \sin^{2} x - \sin^{2} x \ln (\sec x + \tan x)$

Since 1 is in fundamental set solution so $y_{p} = \ln | \sec x | - \sin x \ln | \sec x + \tan x |$

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In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
$y^{'''} + y^{'} = t a n x$

Short Answer

Step by step solution

Definition

Find complementary solution

Calculate Wornkians

For particular solution

One App. One Place for Learning.

Most popular questions from this chapter

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In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.y'''+y'=tanx

Short Answer

Step by step solution

Definition

Find complementary solution

Calculate Wornkians

For particular solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Probability and Statistics

Calculus

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
$y^{'''} + y^{'} = t a n x$