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Chapter 8: Q22P (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 $(6.24),$ .Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see $(6 .26)$ andthe discussion after].
$2 y^{''} + y^{'} = 2 x$

Short Answer

Expert verified

The general solution given by differential equation is $y (x) = C_{1} + C_{2} e^{鈭� \frac{x}{2}} + x^{2} 鈭� 4 x$

Step by step solution

01

Given data.

Given equation is $2 y^{''} + y^{'} = 2 x$

02

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.

03

Find the general solution of given differential equation.2y''+y'=2x

The given differential equation is

$\begin{array}{l} 2 y^{''} + y^{'} = 2 x \\ 2 D^{2} + D = 2 x \end{array}$

The auxiliary equation can be written as

$\begin{matrix} 2 m^{2} + m = 0 \\ m (2 m + 1) = 0 \\ m = 0, 鈭� \frac{1}{2} \end{matrix}$

Solve complementary function and particular integral.

$C . F = C_{1} + C_{2} e^{鈭� \frac{x}{2}}$

$\begin{matrix} P . I = \frac{1}{2 D^{2} + D} 2 x \\ 鈬� \frac{1}{D (2 D + 1)} 2 x \\ 鈬� \frac{1}{D} {(1 + 2 D)}^{- 1} 2 x \\ 鈬� \frac{1}{D} (1 鈭� 2 D) 2 x \end{matrix}$

Solve the equation further

$\begin{matrix} 鈬� \frac{1}{D} (2 x 鈭� 4) \\ 鈬� x^{2} 鈭� 4 x \\ P . I = x^{2} 鈭� 4 x \end{matrix}$

Solution of the differential equation can be written as,

$y (x) = C_{1} + C_{2} e^{鈭� \frac{x}{2}} + x^{2} 鈭� 4 x$

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Most popular questions from this chapter

Obtain $L (t e^{- a t} c o s b t)$

Use L28 and L4 to find the inverse transform of $p e^{- p t} I (p^{2} + 1)$ .

Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.

$y^{"} + 2 y^{'} + 2 y = | x |, - 蟺 < x < 蟺$ .

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y^{'}' + 9 y = \cos 3 t, 鈥� y_{0} = 2, y_{0}^{'} = 0$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y " - 4 y = 4 e^{2 t}, \underset{}{鈭�} y_{0} = 0, y'_{0} = 1$

See all solutions

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