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Chapter 4: Q. 8 (page 402)

Initial-Value Problems: An initial-value problem is a differential equation together with an initial condition that specifies one value of a function.

  • Find the solution f to the initial-value problem f'(x)= f(x),f(0)=3.

Short Answer

Expert verified

Ans: The initial-value problem f'(x)=f(x),f(0)=3.

Step by step solution

01

Step 1. Given information:

The initial-value problem f'(x)=f(x),f(0)=3.

02

Step 2. Solving the differential equation :

In order to solve the given differential equation f'(x)=f(x), observe that the solution will be a function whose derivative is the function itself. Observe that such a function is the exponential function exbecause the derivative is ddxex=ex.

So, one such function that satisfies the given differential equation could be f(x)=Cex.

Now use the initial condition f(0)=3and simplify further for C to get

f(0)=Ce03=C(1)3=C鈬�C=3

Thus, the solution of the given differential equation is f(x)=3ex.

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Most popular questions from this chapter

Fill in each of the blanks:

(a) 鈭�x6dx=+C.

(b) is an antiderivative of role="math" localid="1648619282178" x6.

(c) The derivative of is x6.

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: tanx=sinxcosx).

鈭�2xlnx-xlnx2dx.

If 鈭�-23f(x)dx=4,鈭�-26f(x)dx=9,鈭�-23g(x)dx=2, and 鈭�36g(x)dx=3, then find the values of each definite integral in Exercises 29-40. If there is not enough information, explain why.

鈭�-2-2x(f(x)+3)2dx

Approximate the area between the graph f(x)=x2and the x-axis from x=0 to x=4 by using four rectangles include the picture of the rectangle you are using

Use a sentence to describe what the notation 鈭�k=387k2means. (Hint: Start with 鈥淭he sum of....鈥�)
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