Chapter 4: Q14E (page 191)

In Problems 11–18, find a general solution to the differential equation
$y'' (θ) + y (θ) = se c^{3} θ$ .

Short Answer

Expert verified

The general solution is $y (t) = c_{1} \cos t + c_{2} \sin t - (\frac{1}{2 \sec^{2} θ}) \cos θ + (\tan θ) \sin θ$ .

Step by step solution

Find a particular solution

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

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

Then $y_{1} = \cos θ, v_{2} = \sin θ$

$y_{h} (t) = c_{1} \cos θ + c_{2} \sin θ$

The particular solution is $y_{p} = y_{1} (θ) c o s θ + y_{2} (θ) s i n θ$ .

Evaluate v1 and v2

Here $y_{p} = y_{1} (θ) \cos θ + y_{2} (θ) \sin θ$

Here $a = 1$ and $f = \sec^{3} θ$

And referring to (9) $y (t) = c_{1} e^{α t} \cos β t c_{2} e^{α t} \sin β t$ and solve the system by derivative then;

$\begin{array}{rcl} y_{1}' \cos θ + y_{2}' \sin θ & = & 0 \\ - y_{1}' \sin θ + y_{2} \cos θ & = & \frac{f}{a} \\ - y_{1}' \sin θ + y_{2} \cos θ & = & \sec^{3} θ \end{array}$

Find v1' and v1

$\begin{array}{rcl} v_{1}' & = & \frac{- f (θ) v_{2} (θ)}{a [v_{1} (θ) v'_{2} (θ) - v'_{1} (θ) v_{2} (θ)]} \\ = & \frac{- (\sec^{3} θ) . \sin θ}{[\cos^{2} θ + \sin^{2} θ]} \\ = & - (\sec^{3} θ) . \sin t \end{array}$

Now integrating this;

$\begin{array}{rcl} v_{1} (t) & = & \int - (\sec^{3} θ) . \sin t d t \\ = & - \frac{1}{2 \sec^{2} θ} + C \end{array}$

Determine v2' and v2

$\begin{array}{rcl} v_{2}' & = & \frac{- f (θ) v_{2} (θ)}{a [v_{1} (θ) v'_{2} (θ) - v'_{1} (θ) v_{2} (θ)]} \\ = & \frac{- (\sec^{3} θ) . \cos θ}{[\cos^{2} θ + \sin^{2} θ]} \\ = & (\sec^{2} θ) \end{array}$

Integrate this.

$\begin{array}{rcl} y_{2} (t) & = & \int \sec^{2} θ d t \\ = & \tan θ + C \end{array}$

Thus, the particular solution is:

$y_{p} = (- \frac{1}{2 x^{2}} + C) \cos θ + (\tan θ + C) \sin θ y_{p} = (- \frac{1}{2 \sec^{2} θ}) \cos θ + (\tan θ) \sin θ$

Therefore, the general solution is:

$y (t) = y_{h} (t) + y_{p} (t) y (t) = c_{1} c o s t + c_{2} s i n t - (\frac{1}{2 s e c^{2} θ}) c o s θ + (t a n θ) s i n θ$

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In Problems 11–18, find a general solution to the differential equation
$y'' (θ) + y (θ) = se c^{3} θ$ .

Short Answer

Step by step solution

Find a particular solution

Evaluate v1 and v2

Find v1' and v1

Determine v2' and v2

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In Problems 11–18, find a general solution to the differential equationy''(θ)+y(θ)=sec3θ.

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

Mechanics Maths

Logic and Functions

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 11–18, find a general solution to the differential equation
$y'' (θ) + y (θ) = se c^{3} θ$ .