Chapter 8: Q17P (page 423)

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by $(6.18), (6.23),$ or. $(6 .24)$ Alsofind a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see $(6 .26)$ andthe discussionafter $(6 .15)$ ].
$y^{''} + 16 y = 16 c o s 4 x$

Short Answer

Expert verified

The general solution given by differential equation is

$y (x) = C_{1} \sin 4 x + C_{2} \cos 4 x + 2 x \sin 4 x$

Step by step solution

Given data.

Given equation is $y^{''} + 16 y = 16 \cos 4 x$

General solution of differential equation.

A general solution to the nth order differential equation is one that incorporates a significant number of arbitrary constants. If one uses the variable approach to solve a first-order differential equation, one must insert an arbitrary constant as soon as integration is completed.

Find the general solution of given differential equation. y''+16y=16cos4x

Substitute the value as,

$D = y^{'}, D^{2} = y^{''}$

$\begin{matrix} y^{''} + 16 y = 16 \cos 4 x \\ (D^{2} + 16) y = 16 \cos 4 x \\ \Rightarrow m^{2} + 16 = 0 \\ \Rightarrow m = \pm 4 i \end{matrix}$

$\begin{array}{l} C . F = C_{1} \sin 4 x + C_{2} \cos 4 x \\ P . I = \frac{1}{D^{2} + 16} 16 \cos 4 x \end{array}$

(Putting, $D^{2} = - a^{2}$ denominator becomes 0)

$\Rightarrow \frac{x}{2 D} 16 \cos 4 x$

(Differentiating denominator by D and multiplying numeratorby x )

$\begin{array}{l} \Rightarrow \frac{8 x \sin 4 x}{4} \\ P. I = 2 x \sin 4 x \\ C. S = C_{1} \sin 4 x + C_{2} \cos 4 x + 2 x \sin 4 x \end{array}$

The solution of the differential equation is

$y (x) = C_{1} \sin 4 x + C_{2} \cos 4 x + 2 x \sin 4 x$

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Short Answer

Step by step solution

Given data.

General solution of differential equation.

Find the general solution of given differential equation. y''+16y=16cos4x

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