DE Solution: y' -2y/(x+a) = -1 with Integrating Factor (x+a)^-2

[MACH2]

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??

 

[slider142]

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??

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]

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

The above isn't right. The left side is [y * (x + a)^-2]', or in another form d/dx[y * (x + a)^-2].

y = C(x+a)^2 + (x+a) this seems to be a solution, C = arbitrary constant

Is this the only solution??

 

[MACH2]

The above isn't right. The left side is [y * (x + a)^-2]', or in another form d/dx[y * (x + a)^-2].

(x+a)^-2 = (x+a) elevated to the -2 power (I am not used to this sites equation editor)

Thanks for your reply,

 

[MACH2]

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.

Thank you for the reply.

 

[Mark44]

The above isn't right. The left side is [y * (x + a)^-2]', or in another form d/dx[y * (x + a)^-2].

(x+a)^-2 = (x+a) elevated to the -2 power (I am not used to this sites equation editor)

It was the left side I was talking about, not the right side. Inside the brackets you should not have y'.