Separable Differential Equation, (initial value problem) Quiz solution

TL;DR

Solve differential equation using separation of variables method to find the general solution step by step.

Transcript

okay we are going to solve this ignition for your problem we have D by DX is equal to e to the Y times a X and then Y of zero is equal to zero so let's focus on getting the general solution first well we have dy DX like this so let's multiply 2x on both sides and that's my recommendation you should not have the differential in the denominator then ... Read More

Key Insights

• ❓ Separating variables is a fundamental step in solving differential equations efficiently.
• 🙃 Integrating both sides allows for isolating the variables and finding the final solution.
• 🦻 Initial conditions aid in determining the arbitrary constants present in the solution.
• 😑 Removing negative signs before simplifying logarithmic expressions is essential for valid results.

Questions & Answers

Q: What is the first step in solving the given differential equation?

The first step is to separate the variables by moving all terms involving Y to one side and terms involving X to the other side.

Q: How do you handle the e to the Y term when integrating both sides?

The term e to the Y is dealt with by dividing both sides by e to the Y, followed by further integration to isolate the variables.

Q: Why is it important to remove the negative sign before finding the final solution?

Removing the negative sign is crucial because taking the natural logarithm of a negative number yields complex results, which need to be avoided.

Q: How is the constant C determined in the final solution?

The constant C is determined by applying the initial condition Y of zero = 0 and solving for C, which leads to the final expression for Y as a function of X.

Summary & Key Takeaways

• Given a differential equation D by DX = e to the Y times X and initial condition Y of zero = 0.

• Initial focus on getting the general solution by separating variables and integrating both sides.

• Derive the final solution in the form of Y as a function of X, leading to Y = -Ln(cosine X).