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Chapter 10: Q. B (page 777)

A certain set of parametric equations has the properties that:

x'(t)=x(t),y'(t)=x(t)y(t),x(0)=1andy(0)=e

(a) Solve the system of differential equations. (Hint: Start by solving the initial-value problem x'(t)=x(t)withx(0)=1) (b) Graph the parametric curve defined by x=x(t)andy=y(t)by eliminating the parameter

Short Answer

Expert verified

(a).

(b). The graph is,

Step by step solution

01

Given information

x'(t)=x(t),y'(t)=x(t)y(t),x(0)=1andy(0)=e

02

Part (a) Step 1: Solve

We have,

x'(t)=x(t)鈬�鈭�x'(t)dt=鈭�x(t)dt鈬�x(0)=鈭�x(0)dt+c鈬�1=鈭�1dt+c鈬�1=t+cy'(t)=x(t)y(t)鈬�鈭�y'(t)dt=鈭�x(t)y(t)dt鈬�鈭�y'(t)dt=x(t)鈭�y(t)dt-鈭�x'(t)鈭�y(t)dtdt鈬�y(t)=x(t)鈭�y(t)dt-鈭�x'(t)鈭�y(t)dtdt鈬�y(0)=x(0)鈭�y(0)dt-鈭�x'(0)鈭�y(0)dtdt鈬�e=1鈭�edt-鈭�1鈭�edtdt鈬�e=et-et22+c

03

Part (b) Step 1: Graph

The graph of the function is as shown below;

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