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

Step by step solution

Given data.

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

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.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

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

Solve complementary function and particular integral.

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

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

Solve the equation further

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

Solution of the differential equation can be written as,

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

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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.2y''+y'=2x

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