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2.2.2 Homogeneous Equations

# 2.2.2 Homogeneous Equations - GE 207K Summary Homogeneous...

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GE 207K Homogeneous Equations September 15, 2011 Summary: A di ff erential equation which can be written in the form y = g y x . In other words, the di ff erential equation can be written such that anywhere the dependant variable appears, it appears as fraction y x . These equations can be turned into separable equations by performing a substitution. These homogenous equations are di ff erent than the class of di ff erential equations which we classified as homogenous on the first day. The following steps can be taken to find the general solution to the di ff erential equation: Steps: 1. Write the given di ff erential equation in the right form. y = g y x (1) 2. Eliminate y from the di ff erential equation entirely by rewriting the di ff erential equation in terms of x and (a newly introduced variable) u . Do this by performing the following substitutions: (a) Perform the substitution in the di ff erential equation u = y x . (2) (b) Replace y with the following expression y = ux, y = u x + u (chain rule) . (3) 3. If you’ve done everything right, the di ff erential equation should be a separable equation at this point (in terms of u and x only). Solve by separating variables and integrating both sides of the resulting equation.

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- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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2.2.2 Homogeneous Equations

# 2.2.2 Homogeneous Equations - GE 207K Summary Homogeneous...

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GE 207K Homogeneous Equations September 15, 2011 Summary: A di ff erential equation which can be written in the form y = g y x . In other words, the di ff erential equation can be written such that anywhere the dependant variable appears, it appears as fraction y x . These equations can be turned into separable equations by performing a substitution. These homogenous equations are di ff erent than the class of di ff erential equations which we classified as homogenous on the first day. The following steps can be taken to find the general solution to the di ff erential equation: Steps: 1. Write the given di ff erential equation in the right form. y = g y x (1) 2. Eliminate y from the di ff erential equation entirely by rewriting the di ff erential equation in terms of x and (a newly introduced variable) u . Do this by performing the following substitutions: (a) Perform the substitution in the di ff erential equation u = y x . (2) (b) Replace y with the following expression y = ux, y = u x + u (chain rule) . (3) 3. If you’ve done everything right, the di ff erential equation should be a separable equation at this point (in terms of u and x only). Solve by separating variables and integrating both sides of the resulting equation.

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### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern
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