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Chapter 1: Q2 E (page 1)

The direction field for dydx=2x+yas shown in figure 1.13.

  1. Sketch the solution curve that passes through (0, -2). From this sketch, write the equation for the solution.

b. Sketch the solution curve that passes through (-1, 3).

c. What can you say about the solution in part (b) as x鈫�+鈭�? How about x鈫�-鈭�?

Short Answer

Expert verified
  1. The graph is drawn below, and the equation is y=-2-2x.
  2. The graph is drawn below, and the equation isy=1-2x.
  3. The solutions become infinite whenx鈫�+鈭�orx鈫�-鈭�.

Step by step solution

01

Step 1(a): Find the curve by point (0,-2)

Givendydx=2x+y鈥�......(1)

Put the value of the point (0,-2) in equation (1)

m=2(0)-2=-2

The curve is(y+2)=-2(x-0)

Hence the solution is y=-2-2x.

By putting the different values of x, get the values of y.

02

Step 2(b): Find the curve by point (-1,3).

Givendydx=2x+y.鈥�.....(2)

Put the value of the point (-1,3) in equation (2)

m=2(-1)-3=1

The curve isy=1-2x

Hence the solution is y=1-2x.

03

Step 3(c): Discuss the solution in part (b) as x→+∞ and x→-∞

As x鈫�鈭�the solution becomes infinite. And whenx鈫�-鈭�the solution also becomes infinite and has an asymptotey=-2-2x.

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