First order homogenous equations | First order differential equations | Khan Academy

TL;DR

Homogeneous differential equations can be solved by making a variable substitution, turning them into separable equations.

Transcript

I will now introduce you to the idea of a homogeneous differential equation. Homogeneous is the same word that we use for milk, when we say that the milk has been-- that all the fat clumps have been spread out. But the application here, at least I don't see the connection. Homogeneous differential equation. And even within differential equations, w... Read More

Key Insights

• 💄 Homogeneous differential equations can be solved by making a substitution that transforms them into separable equations.
• ❣️ The substitution involves setting v = y/x, where v represents the ratio of y to x.
• 😰 By finding the derivative of v with respect to x and integrating, the solution to the homogeneous equation can be obtained.
• 😑 The solution to a homogeneous differential equation can be expressed in terms of the original variables (y and x) by reversing the substitution made during the solution process.
• 😀 Initial conditions can be used to determine the constant (c) in the solution for a particular homogeneous differential equation.
• 🤩 Algebraic manipulation is a key step in recognizing and solving homogeneous differential equations.
• 🔨 Homogeneous differential equations provide a useful tool for solving certain types of differential equations that are not easily separable or exact.

Questions & Answers

Q: What is the difference between a regular first order differential equation and a homogeneous one?

In a regular equation, the function on the right side can be any function of x and y. In a homogeneous equation, the function on the right side can be rewritten as a function of y/x.

Q: How do you make a variable substitution for y/x?

To make the substitution, let v = y/x. This allows you to rewrite dy/dx and substitute it back into the equation to solve for v.

Q: Can you provide an example of solving a homogeneous differential equation?

Sure! Let's take the example dy/dx = x + y/x. By dividing the top terms by x, we get dy/dx = 1 + y/x. Making the substitution v = y/x, we can solve the equation as x dv/dx = 1, integrate, and reverse substitute to find the solution.

Q: How can you verify the solution for a homogeneous differential equation?

You can verify the solution by substituting the derived equation back into the original differential equation and confirming that it satisfies the equation.

Summary & Key Takeaways

• Homogeneous differential equations can be made separable by algebraically manipulating the right side of the equation.

• By dividing the top terms by x and rewriting the equation, it becomes a function of y/x.

• Making a substitution for y/x (v = y/x) allows the equation to be solved by finding the derivative of v with respect to x and integrating.