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eightlgddj
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Hi, I'm new to the forum, and new to differential equations. I was wondering if someone could post a no-nonsence explanation of substitution methods for first order differential equations.
Thanks!
Thanks!
eightlgddj said:I mean one like this:
saltydog said:That equation is homogeneous of degree 2. Thus, you can make a substitution y=vx, and then separate variables. Your ODE book should have this technique as a separate section at the beginning unless you have one of those "qualitative books" like Devaney's. I'm old-school.
A substitution method is a technique used to solve differential equations by replacing the dependent variable with a new variable. This new variable is chosen in a way that simplifies the equation and makes it easier to solve.
A substitution method is typically used when the differential equation is not in a form that can be solved directly, such as when it is not separable or linear. It is also useful when the equation contains multiple variables or functions.
Some common substitution methods include the change of variables method, the substitution of a power series, and the Laplace transform method. Each method has its own advantages and may be more suitable for certain types of differential equations.
No, not all 1st order differential equations can be solved using a substitution method. Some equations may require more advanced techniques or may not have analytical solutions at all.
One tip is to carefully choose the substitution variable to simplify the equation as much as possible. It may also be helpful to practice with simpler equations before moving on to more complex ones. Additionally, double-checking the solution by plugging it back into the original equation can help ensure accuracy.