Sketch the solution curves for y'=xy
How do I sketch this??? Where do I start?
To sketch the solution curves for the given differential equation y' = xy, you can follow these steps:
1. Start by finding some specific solutions to the equation. These are known as "particular solutions." To do this, you will integrate both sides of the equation.
∫ (1/y) dy = ∫ x dx
ln|y| = (1/2)x^2 + C
|y| = e^((1/2)x^2 + C)
|y| = e^C * e^((1/2)x^2)
|y| = k * e^((1/2)x^2), where k = ± e^C
2. Notice that the equation includes the absolute value of y, which means the graph could be symmetric about the y-axis. To simplify, drop the absolute value by considering both positive and negative values of k.
y = ± k * e^((1/2)x^2)
3. Choose values for k, and plot the corresponding curves on your coordinate axes. Since k can take any real value, you will find infinitely many unique curves passing through different points in the xy-plane.
4. It's helpful to plot a few specific solution curves, such as when k = 1, k = 2, and k = -1. This will give you an idea of what the curves look like.
- For k = 1, y = e^((1/2)x^2)
- For k = 2, y = 2e^((1/2)x^2)
- For k = -1, y = -e^((1/2)x^2)
5. Finally, sketch the curves that you obtained in step 4 on your coordinate axes. You should observe a family of curves that are symmetric and seem to grow/shrink exponentially as x increases.
Remember that the given steps outline a general approach, and it's always useful to manipulate the equation and plot specific cases to gain a better understanding of the curves.