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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 yp is,

Substitute the values of yp and yc 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

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from y'for the original curves; this constant takes different values for different curves of the original family, and you want an expression for y'which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations (2.10)to (2.12)

x2+y2=cost.

In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.

13. Problem 2

Verify the statement of Example 2. Also verify that y=coshxandy=sinhx are solutions of y''=y.

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

y篓+4y+5y=26e3t,藱y0=1,y0=5藱

Show that鈭�-鈭�鈭�fn(t)dt=1for the functionsfn(t)in Figures 11.3 and 11.4.
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