Chapter 4: Q2E (page 191)

In Problems 1–8, find a general solution to the differential equation using the method of variation of parameters. $y'' + y = sect$

Short Answer

Expert verified

The general solution is $y (t) = c_{1} cost + c_{2} sint + (\ln | cost |) cost + tsint$ .

Step by step solution

Find a particular solution.

The homogenous equation is $r^{2} + 1 = 0$ .

Two independent solutions are $r = \pm i$ .

$y_{1} = cost, y_{2} = sint$

Then $y_{h} (t) = c_{1} cost + c_{2} sint$

The particular solution is $y_{p} = v_{1} (t) cost + v_{2} (t) sint$

evaluate v1 and v2.

Here $y_{p} = v_{1} (t) cost + v_{2} (t) sint$

And referring to (9) and solve the system then

role="math" localid="1655137296914" $\begin{matrix} v_{1}' cost + v_{2}' sint = 0 \\ - v_{1}' sint + v_{2}' cost = \frac{f}{a} \\ - v_{1}' sint + v_{2}' cost = sect \end{matrix}$

Find v1' and v1.

$\begin{matrix} v_{1}' = \frac{- f (t) y_{2} (t)}{a [y_{1} (t) y'_{2} (t) - y'_{1} (t) y_{2} (t)]} \\ = \frac{- sect . sint}{1 [- \cos^{2} t . - \sin^{2} t]} \\ = \frac{- sect . sint}{[- \cos^{2} t - \sin^{2} t]} \\ = tant \end{matrix}$

Now integrating this.

$\begin{array}{l} v_{1} (t) = \int - tant dt \\ = \ln | cost | + C \end{array}$

Determine v2' and v2 .

$\begin{matrix} v_{2}' = \frac{f (t) y_{1} (t)}{a [y_{1} (t) y'_{2} (t) - y'_{1} (t) y_{2} (t)]} \\ = \frac{sect . cost}{1 [\cos^{2} t + \sin^{2} t]} \\ = 1 \end{matrix}$

Integrate this.

$\begin{array}{l} v_{2} (t) = \int 1 dt \\ = t + C \end{array}$

Thus particular solution is

$\begin{matrix} y_{p} = (\ln | cost | + C) cost + (t + C) sint \\ y_{p} = (\ln | cost |) cost + tsint \end{matrix}$

Thus, general solution is:

$\begin{matrix} y (t) = y_{h} (t) + y_{p} (t) \\ y (t) = c_{1} cost + c_{2} sint + (\ln | cost |) cost + tsint \end{matrix}$

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In Problems 1–8, find a general solution to the differential equation using the method of variation of parameters. $y'' + y = sect$

Short Answer

Step by step solution

Find a particular solution.

evaluate v1 and v2.

Find v1' and v1.

Determine v2' and v2 .

One App. One Place for Learning.

Most popular questions from this chapter

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In Problems 1–8, find a general solution to the differential equation using the method of variation of parameters. y''+y=sect

Short Answer

Step by step solution

Find a particular solution.

evaluate v1 and v2.

Find v1' and v1.

Determine v2' and v2 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Logic and Functions

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 1–8, find a general solution to the differential equation using the method of variation of parameters. $y'' + y = sect$