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Chapter 1: Q18E (page 13)

In Problemsverify that the indicated functionis an explicit solution of the given first-order differential equation. Proceed as in Example, by consideringsimply as a function and give its domain. Then by consideringas a solution of the differential equation, give at least one intervalof definition.

Short Answer

Expert verified

Answer:

The indicated function is an explicit solution of the given differential equation and the interval is , and the domain is .

Step by step solution

01

Step by Step solution:Step 1: Define an explicit function.

Inexplicit solutionunknown functions (dependable variables) is indicated (expressed) in terms of undependable variable and constants. This is known as “explicit solution”.

Let the given function be.

Then, the first derivative of the function is,

02

Determine the explicit solution.

Substituteandin the left-hand side of the differential equation.

That is same as the right-hand side of the differential equation. The indicated function is an explicit solution of the given differential equation.

03

Determine the domain and the interval.

Hence the domain of the solution while considering the solution as a function is,

And the interval is.

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Most popular questions from this chapter

In Problems, 17-24 determine a region of the xy-plane for which

the given differential equation would have a unique solution whose

graph passes through a point(x0,y0) in the region.

dydx=y23

Determine whether the given differential equation is exact. If it is exact, solve it.

(1+lnx+yx)dx=(1-lnx)dy

In Problems 35–38 the graph of a member of a family of solutions of a second-order differential equationd2y/dx2=f(x,y,y')is given. Match the solution curve with at least one pair of the following initial conditions.

a)y(1)=1,y'(1)=−2b)y(−1)=0,y'(−1)=−4c)y(1)=1,y'(1)=2d)y(0)=−1,y'(0)=2e)y(0)=−1,y'(0)=0f)y(0)=−4,y'(0)=−2

FIGURE 1.2.8 Graph for Problem 36

In Problems11-14, 11-14,y=c1ex+c2e-xis a two-parameter family of solutions of the second-order DEy''-y=0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

y(0)=0,y'(0)=0

In Problems 3 and 4 Fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1and c2and has the form F(y',y")=0. The symbols c1, c2, and k represent constants.

d2dx2(c1coskx+c2sinkx)=________
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