Yes, your 1-parameter family of solutions contains every particular solution. In fact, every equation of this form, when solved by multiplying by the integrating factor of the exponential of the integral of p, will yield a 1-parameter family of solutions that contains every particular solution, as long as p and q are continuous on the interval of the solution.MACH2 said:Homework Statement
Solve DE: y' -2 y / (x + a) = -1 where a = constant
Homework Equations
y' + p(x) y = q(x) solving this DE with integrating factor.
The Attempt at a Solution
Use integrating factor (x+a) ^ -2 for above DE,
[ y'(x+a) ^ -2 ]' = - (x+a)^ -2 solving this DE we get
y = C(x+a)^2 + (x+a) this seems to be a solution, C = arbitrary constant
Is this the only solution??
The above isn't right. The left side is [y * (x + a)-2]', or in another form d/dx[y * (x + a)-2].MACH2 said:Homework Statement
Solve DE: y' -2 y / (x + a) = -1 where a = constant
Homework Equations
y' + p(x) y = q(x) solving this DE with integrating factor.
The Attempt at a Solution
Use integrating factor (x+a) ^ -2 for above DE,
[ y'(x+a) ^ -2 ]' = - (x+a)^ -2 solving this DE we get
MACH2 said:y = C(x+a)^2 + (x+a) this seems to be a solution, C = arbitrary constant
Is this the only solution??
Mark44 said:The above isn't right. The left side is [y * (x + a)-2]', or in another form d/dx[y * (x + a)-2].
slider142 said:Yes, your 1-parameter family of solutions contains every particular solution. In fact, every equation of this form, when solved by multiplying by the integrating factor of the exponential of the integral of p, will yield a 1-parameter family of solutions that contains every particular solution, as long as p and q are continuous on the interval of the solution.
Mark44 said:The above isn't right. The left side is [y * (x + a)-2]', or in another form d/dx[y * (x + a)-2].
It was the left side I was talking about, not the right side. Inside the brackets you should not have y'.MACH2 said:(x+a)^-2 = (x+a) elevated to the -2 power (I am not used to this sites equation editor)
A "Re-post Solution to DE" refers to the process of sharing or reposting a solution to a differential equation (DE). This can be done on various platforms, such as online forums or social media, in order to help others who may be struggling with the same problem.
There are a few reasons why someone may need a "Re-post Solution to DE." It could be because they are stuck on a particular step or concept in solving the DE, or they may want to compare their solution to others' to check for any mistakes. Additionally, some individuals may simply want to learn different methods or approaches to solving the same DE.
There are several ways to find a "Re-post Solution to DE." One option is to search for the specific DE on online forums or social media platforms, where people often share their solutions and ask for feedback. Another option is to look for resources, such as textbooks or websites, that provide step-by-step solutions to common DEs.
It depends on the source of the solution. If the solution comes from a reputable source, such as a textbook or a trusted expert, then it is likely trustworthy. However, if the solution is shared by an individual on a forum or social media platform, it is always a good idea to double-check the steps and make sure they are correct.
Using a "Re-post Solution to DE" can be ethical as long as it is used for learning purposes only. It is important to understand the steps and concepts behind the solution rather than just copying it. Additionally, if the solution is from a source that requires proper citation, it is important to give credit to the original author.