Solving Differential Equations: An Analysis of dy/dt y^21/t1 with Initial Conditions

Introduction

In this article, we will analyze and solve a differential equation that often arises in various mathematical and physical contexts. Specifically, we are going to consider the differential equation y21yt, where the initial condition is given as y40whent18. This type of equation is a separable differential equation, and we will demonstrate the steps to solve it.

Separeting the Variables

To solve this differential equation, we need to separate the variables. Start by rearranging the equation so that all terms involving y are on one side and all terms involving t are on the other side:

dyydtty21

Next, integrate both sides:

∫C1C21y21dy∫C3C41tdt

The integral of 1y21 with respect to y is:

y→?11-21-120y-20

The integral of 1t with respect to t is:

t→?lnt

Therefore, we have:

-120y-20lnt C

Solving for y

To solve for y, we first isolate y on one side of the equation:

y 20?-1(-20×lnt 20×C)1_pipe20?-1-20

Simplifying it gives:

ytan?1lnt C

Applying the Initial Conditions

We are given that when t18, y40. Substitute these values into the equation to solve for the constant C:

40tan?1ln18 C

Calculating the arctan of both sides:

Ctan：?140?ln18?1.3986374

Thus, the solution is:

ytan?1lnt ?1.3986374

Periodic Behavior

Since the tangent function is periodic with a period of π, the constant C can be expressed as C?1.3986374 πn, where n is an integer.

Conclusion

In conclusion, we have successfully solved the given differential equation with the specified initial condition. The solution is:

ytan?1lnt ?1.3986374

where C is the constant of integration that is derived from the initial condition and can be adjusted based on periodic behavior.

References

1. Boyce, W. E., DiPrima, R. C. (1986). Elementary Differential Equations and Boundary Value Problems. Wiley.

2. Zill, D. G., Wright, W. S. (2012). Differential Equations with Boundary-Value Problems. Cengage Learning.