Homogeneous Differential Equations
The Organic Chemistry Tutor・2 minutes read
Homogeneous differential equations involve terms of the same degree and can be solved by substituting y with v times x, then integrating and simplifying the equation using properties of logarithms. The final solution for the homogeneous differential equation involves the absolute value of x equaling c times x minus y squared, with the general solution found to be e to the y over x equals ln x squared plus c, and the constant value determined by using the initial point (1, 0) to be c equals 1.
Insights
- Homogeneous differential equations involve terms of the same degree, and solving them requires a systematic approach of substitution, integration, and constant determination.
- The general solution for a homogeneous differential equation can be simplified through steps like replacing variables, integrating, and determining constants, leading to a final answer that encapsulates the relationship between x and y.
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Recent questions
How are homogeneous differential equations solved?
By replacing y with v times x and integrating.
What is the final solution for a homogeneous differential equation?
The absolute value of x equals c times x minus y squared.
How is the constant value determined in solving a differential equation?
By using the initial point to find the value.
What role does substitution play in solving differential equations?
It simplifies the equation and aids in integration.
How are logarithmic properties utilized in solving differential equations?
To simplify expressions and arrive at the final solution.
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Summary
00:00
Solving Homogeneous Differential Equations Step-by-Step
- Homogeneous differential equations have terms of the same degree.
- To solve a homogeneous differential equation, replace y with v times x and differentiate the function.
- Replace dy with x dv plus v dx in the equation.
- Simplify the equation by factoring and distributing terms.
- Separate variables by dividing both sides by x and v to the third.
- Integrate both sides of the equation.
- Use properties of logarithms to simplify the solution.
- Replace v with y over x to further simplify the solution.
- Condense logarithmic expressions and solve for the final answer.
- The final solution for the homogeneous differential equation is the absolute value of x equals c times x minus y squared.
20:01
Solving Homogeneous Differential Equations with Substitution
- The differential equation is solved by replacing y prime with d y over dx and then replacing y with v times x. After separating the variables and integrating, the general solution is found to be e raised to the y over x equals ln x squared plus c.
- To determine the value of c, the initial point (1, 0) is used, leading to the conclusion that c equals 1. Substituting this value back into the general solution gives the final answer: e to the y over x equals ln x squared plus 1, or equivalently, 1 plus ln x squared.
- The process outlined in the video provides a clear method for solving homogeneous differential equations, emphasizing the steps of substitution, separation of variables, integration, and determining the constant value to arrive at the final solution.
