Solve dy/dx=x*sqrt(y-1)

TL;DR

We solve a differential equation by separating the variables and integrating, using the given initial condition.

Transcript

okay we are going to solve this differential equation along with this initial condition this is not that bad because we can just separate the variables let me show you first of all let's multiply DX on both sides so we will have a DX right here and you want to keep the X right here and I would like to bring the square root of Y minus 1 to the other... Read More

Key Insights

• ❓ Differential equations can be solved by separating the variables and using integration.
• ❓ Initial conditions are crucial in finding the values of the constants and obtaining the specific solution.
• 👻 Dividing by the square root of Y minus 1 allows for easier integration and manipulation of the equation.

Questions & Answers

Q: How do we solve the differential equation with initial conditions?

We begin by separating the variables and integrating to get Y minus 1 raised to the negative 1/2 power. Then, we can solve for the constants by plugging in the given initial conditions.

Q: Why do we divide by the square root of Y minus 1?

Dividing both sides of the equation by the square root of Y minus 1 allows us to isolate Y and X on one side, making it easier to integrate later on.

Q: How do we find the constant C2 using the initial condition?

We can plug in the given X and Y values into the equation and solve for C2. This helps us determine the specific solution that satisfies the initial condition.

Q: What is the final solution for Y as a function of X?

The final solution is Y = 1 + (1/4x^2) + 2, which is obtained by squaring both sides of the equation after finding the values of the constants.

Summary & Key Takeaways

• The differential equation is separated by multiplying by dx and dividing by the square root of Y minus 1.

• Integrating both sides gives the equation Y minus 1 raised to the negative 1/2 power.

• Solving for the constants using the given initial condition leads to the final solution of Y as a function of X.