Question

A differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. $$ \frac{d y}{d x}=x y, \quad\left(0, \frac{1}{2}\right) $$

Short answer

The particular solution of the provided differential equation with the point (0, 1/2) is \(y(x) = e^{\frac{1}{2}x^2 + ln(1/2)}\). This solution sketch on the slope field should closely match the original sketched curves.

Step-by-step solution

Sketch the slope field and solutions

Begin by sketching the slope field of the differential equation \(\frac{dy}{dx} = xy \), using a graphing utility or manually if preferred. This helps in visualizing the general behavior of the solutions. Then, draw two curves that follow the direction of the field, one of them passing through the point \( (0, \frac{1}{2}) \).

Solve the differential equation analytically

Transform the differential equation \(\frac{dy}{dx} = xy \) into the separable form \(\frac{1}{y} dy = x dx \). Afterwards, integrate both sides of this equation to find the general solution. The integral of \(\frac{1}{y} dy \) is \(ln|y| + C_1 \), and the integral of \(x dx\) is \( \frac{1}{2}x^2 + C_2 \). Combining these gives: \(ln|y| = \frac{1}{2}x^2 + C\). Now, to find the particular solution, apply the initial condition \(y(0) = \frac{1}{2}\) to the general solution. Substituting these values gives \(ln| \frac{1}{2}| = C\). Therefore, \(C = ln| \frac{1}{2}|\) and our particular solution is \(y(x) = e^{\frac{1}{2}x^2 + ln| \frac{1}{2}|}\).

Graph the particular solution and compare

With a graphing utility, sketch the graph of the particular solution \(y(x) = e^{\frac{1}{2}x^2 + ln| \frac{1}{2}|}\) found in the previous step. Compare this graph to the manually drawn approximate solutions from the first step. If done correctly, they should closely match.