Given the differential equation dy/dx=1/xy

Question

a.) Sketch the slope field for the points (1,+or- 1), (2,+or-1), and (-1,+or-1).

b.) Find the general solution of the given differential equation.

c) Find the solution of this differential equation that satisfies the initial condition y(1)=2.

Answer 1

Here is a graph that shows the slope field for the dy/dx = 1/xy. The specified points in part a) are highlighted in red.

desmos.com/calculator/qc84ept5ys

To find the general solution, separate the variables.

ydy = dx/x, then integrate both sides:

(1/2)y² = ln(|x| + K

To solve for y, multiply both sides by 2, then take square root of both sides:

y² = 2ln|x| + C (2K is just a constant, so replace with C)

y = ±√[ln(x²) + C] is the general solution

Note that x ≠ 0.

For the particular solution at (1,2), the y value is positive so use the positive square root.

2 = √[ln(1)+C]

4 = C

Particular solution:

y = √[ln(x²) + 4]. Since x = 1 is on the branch where x > 0, this particular solution has x > 0.

This Desmos graph shows the particular solution along with the tangent line at the point (1,2).

desmos.com/calculator/qmfpzozllm