Q. 61 - Calculus

Question

Sketch slope fields for each of the differential equations in Exercises $59-64$, and within each slope field sketch four different approximate solutions of the differential equation.

$\frac{dy}{dx}=x+y$.

Step-by-Step Solution

Answer

A graph for the slope field for the differential equation $\frac{dy}{dx}=x+y$ is,

Step 1 . Given information

$\frac{dy}{dx}=x+y$.

Step 2 . The slope field of a differential equation d y d x = g x , y comprises of line segments whose slope at any point a , b is given by d y d x a , b ,

In the present case, the differential equation $\frac{dy}{dx}=x+y$ consists of the line segments whose slope at any point $\left(a,b\right)$ is equal to $a+b$.

Step 3 . Draw the slope field of the differential equation as shown below.

Use different colors to mark the different solutions.

Step 4 . Check the answer.

Note that the given differential equation cannot be solved by either antidifferentiation method or by variable separable method, so its solution cannot be obtained by the known methods for solving differential equation of this chapter. However, the differential equation is of a special type known as linear differential equation. Its solution as solved by the method exclusively designed for this type of differential equation is given as,

$y=-\left(x+1\right)+C{e}^x$

Note that the graph of above solution will be a straight line for $C=0$ and exponential curves for other values of $C$. Observe that the sketches drawn in the field are all curves drawn for $C=-3,0,1,2$. Hence, the approximate solutions are verified to be correct.