implicit differential equations

Question

Given the differential equation dy = e^x · sin( y), verify that a given implicit solution dx

satisfies the ODE. Then, find at least one explicit solution y = φ(x) and use a graphing utility to visualize it.

Part A: (10 pts) Verify that an implicit solution of the form F(x , y) = C, where F(x , y) involves an integration of e^x · sin( y), satisfies the differential equation.

Part B: (10 pts) Find an explicit solution y = φ(x) for a specific initial condition, such as y(0) = 4π . Note: If the equation is not easily solvable for y explicitly, discuss the challenges involved.

Part C: (10 pts) Use a graphing utility to graph the solution, whether implicit or parameterized if explicit is not feasible. Analyze the behavior of the solution over an interval

I and discuss any interesting characteristics or patterns observed.

I have the PDF if anything i can send just please reach out to me asap

Answer 1

A) If we have

dy/dx = e^x * sin(y)

we can separate the variables and then integrate to get an implicit solution. Doing that gives

dy/sin(y) = e^xdx

Integrating gives us

ln |csc(y) - cot(y)| = e^x + C

Verifying that the solution exists just means doing the opposite of what we just did by taking the derivatives of both sides and seeing if we get y' = e^x * sin(y), which in this case will give us that answer.

B) Notice that if we plug in x = 0 and y = 4π to the implicit solution, we will get undefined answers.

The trick with this equation is that it is difficult to solve for y when there are two different trigonometric functions inside on a natural logarithm. It would be best solvable by graph, but here is an explicit solution that is valid.

y(x) = 2cot^-1(e^(C - e^x))

C)

y' = e^x * sin(y) - Wolfram|Alpha (wolframalpha.com)

Here's a vector field of the differential equation. A solution can be made for any x or y value by tracing a line following the vectors of the field. From this you can pick an interval, find the value of C in the explicit form, and impose the solution graph on to the vector field.