Q2 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

The direction field for $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}$ as 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 $\mathbf{x}\mathbf{\to }\mathbf{+}\mathbf{\infty }$? How about $\mathbf{x}\mathbf{\to }\mathbf{-}\mathbf{\infty }$?

Step-by-Step Solution

Answer

1. The graph is drawn below, and the equation is $\mathrm{y}=-2-2\mathrm{x}$.
2. The graph is drawn below, and the equation is $\mathrm{y}=1-2\mathrm{x}$.
3. The solutions become infinite when $\mathrm{x}\to +\infty \mathrm{or} \mathrm{x}\to -\infty$.

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

Given $\frac{\mathrm{dy}}{\mathrm{dx}}=2\mathrm{x}+\mathrm{y} ......\left(1\right)$

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

$\mathrm{m}=2\left(0\right)-2=-2$

The curve is $(\mathrm{y}+2)=-2(\mathrm{x}-0)$

Hence the solution is $\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{-}\mathbf{2}\mathbf{x}$.

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

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

Given $\frac{\mathrm{dy}}{\mathrm{dx}}=2\mathrm{x}+\mathrm{y}. .....\left(2\right)$

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

$\mathrm{m}=2(-1)-3=1$

The curve is $\mathrm{y}=1-2\mathrm{x}$

Hence the solution is $\mathbf{y}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{x}$.

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

As $\mathbf{x}\mathbf{\to }\mathbf{\infty }$ the solution becomes infinite. And when $\mathbf{x}\mathbf{\to }\mathbf{-}\mathbf{\infty }$the solution also becomes infinite and has an asymptote $\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{-}\mathbf{2}\mathbf{x}$.