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Chapter 8: Q 13-27MP (page 466)

Question: In Problems 25to 28, find a particular solution satisfying the given conditions.
when x = 0.

Short Answer

Expert verified

The particular solution for the given differential equation is

.

Step by step solution

01

Given information.

The differential equation is and x = 0.

02

Differential equation.

When fand its derivatives are inserted into the equation, a solution is a function y = f(x) that solves the differential equation. The highest order of any derivative of the unknown function appearing in the equation is the order of a differential equation.

03

Find the solution of the given differential equation.

Consider the equation.

The above equation is a non-homogenous equation.

The auxiliary equation is,

The solution for m = 2, -3 is,

The solution of equation is,

The value of y_p is,

Substitute the values of y_p and y_c in Equation (1).

Differentiate above equation.

Substitute 1 for y and 0 for x in Equation (2).

Substitute 4 for y' and 0 for x in Equation (2).

From Equation (4) and Equation (5).

c₁ = 2

c₂ = 0

Substitute 2 for c₁and 0 for c₂ in Equation (2).

Thus, the particular solution for the given differential equation is.

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Most popular questions from this chapter

$y d y + (x y^{2} - 8 x) d x = 0, y = 3$ when $x = 1$ .

Solve the differential equation $y y^{2} + 2 x y' - y = 0$ by changing from variables role="math" localid="1655272385100" $y, x$ to $r, x$ where $y^{2} = r^{2}$ ; then $y y' = n' - x$ .

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.

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

Using Problems 29 and 31b, show that equation (6.24) is correct.

Several Terms on the Right-Hand Side: Principle of Superposition So far we have brushed over a question which may have occurred to you: What do we do if there are several terms on the right-hand side of the equation involving different exponentials?

In Problem 33 to 38 , solve the given differential equations by using the principle of superposition [see the solution of equation (6.29) . For example, in Problem 33 , solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus a polynomial of any degree is kept together in one bracket.

$y^{"} - 5 y^{'} + 6 y = 2 e^{x} + 6 x - 5$

$(1 + y^{2}) d x + x y d y = 0, y = 0$ when $x = 5$ .

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