How to Find the Equation Relating x and y in Differential Equations

TL;DR

To find the equation relating x and y in a differential equation problem, start with the rate of change given as dy/dx = 1/3y. Integrate both sides and use the provided values to determine the constant C, leading to the final equation log(y/4) = (1/3)x + 1 that describes the connectivity between x and y.

Transcript

click the Bell icon to get latest videos from equator hello friends in this video we are going to see problems which are based on application of rates in differential equations so let us start with problem number 1 the rate of change of a function y with respect to x equals 1 by 3 by + y is equal to 4 when x is equal to minus 1 find the law connect... Read More

Key Insights

• 💱 The rate of change of a function (dy/dx) indicates how the function is changing with respect to another variable (x).
• 💱 To find the equation connecting x and y in a differential equation problem, the given rate of change must be integrated.
• 😀 The general solution includes a constant term (C) that can be determined by substituting the given values into the equation.
• 😑 The final equation connecting x and y can be derived after finding the value of C and simplifying the expression.
• ☠️ Differential equations involve finding the relationships between variables based on their rates of change.
• ❓ The properties of logarithms can be utilized when solving differential equation problems.
• ❓ Integration is an important step in solving differential equations to find the general solution.

Questions & Answers

Q: What does it mean when we say a function is changing with respect to another variable?

When we say a function is changing with respect to another variable, it means that the value of the function (y) is changing as the value of the other variable (x) changes.

Q: How do we find the general solution in a differential equation problem like this?

To find the general solution, we separate the variables and integrate both sides of the equation. In this case, we have 1/y dy = 1/3 dx, which integrates to log(y) = (1/3)x + C.

Q: How do we find the value of C in the general solution?

We can find the value of C by substituting the given values of x and y into the equation. In this problem, when x = -1 and y = 4, we can solve for C to be 4 + 1/3.

Q: What is the final equation that relates x and y in this problem?

The final equation is log(y/4) = (1/3)x + 1. This equation represents the connectivity between x and y based on the given rate of change in the problem.

Summary & Key Takeaways

• This video discusses a problem involving the rate of change of a function in a differential equation and how to find the equation connecting x and y.

• The rate of change, dy/dx, is given as 1/3y, and the goal is to find the law connecting x and y.

• By solving the equation and substituting the given values, the final equation is derived as log(y/4) = (1/3)x + 1.