Question

Sketch a direction field for the differential equation. Then use it to sketch three solution curves.

Short answer

The direction field for the differential equation is drawn and its three solution curves are plotted.

Step-by-step solution

Given

The differential equation is given below,

---(1)

Calculation

Calculate the slopes at several points for the given differential equation,

Substitute 0 for \(x\) and 0 for \(y\) in equation(1),

Similarly, substitute \(1\) for \(x\) and \(1\) for \(y\) in equation(1),

The slopes at several points \(\left( {x,y} \right)\) are given below.

i)

ii)

iii)

iv)

v)

vi)

vii)

viii)

ix)

x)

xi)

xii)

xiii)

xiv)

xv)

Now, draw small line segments with the slopes at their corresponding points \(\left( {x,y} \right)\). The resultant is the direction field for the given differential equation.

Direction field

The direction field for the differential equation

is shown in the below figure.

Now, draw the solution curves which passes through the point by

following the direction field.

Similarly, draw the solution curves which passes through the points

and .

Here, \(a\),\(b\),\(c\), are the three solution curves that passes through their corresponding points (0,0), (0,1),(0, -1) as shown above.

Thus, the direction field for the differential equation

is drawn and its three solution curves are plotted.