How many equations are needed to solve this system?

[enceladus_]

This system originates from systems of linear Diff Eqs. I am required to use the method of undetermined coefficients.

-a2 + 5b2 - a1 = 0
-a2 + b2 -b1 -2 = 0
-a1 + 5b1 +a2 + 1 = 0
-a1 + b1 + b2 = 0

I can't figure out how I am able to solve this. In each case, I have two equations and three unknowns. I know the system is correct.

The answers are:

a2 = -3, b2 = -2/3, a1 = -1/3, b1 = 1/3.

Thanks in advance.

 

[junaid314159]

You can solve your third equation directly:

-a1 + 5b1 +a1 + 1 = 0 → 5b1 + 1 = 0 → b1 = -1/5

This contradicts the answer for b1 you stated. There is either a mistake in the system or in the solution.

Junaid Mansuri

 

[enceladus_]

It should be:

-a1 +5b1 + a2 + 1 = 0

My mistake.

 

[junaid314159]

An easy approach to solve this system is through substitution. Solve for a1 in equation 4 (in terms of the b's). Solve for a2 in equation 2 (in terms of the b's). Substitute the expressions for a1 and a2 into equations 1 and 3 and you will then have a system of two equations with only b1 and b2 in them which is easy to solve. Then you can use the known values of b1 and b2 to directly find a1 and a2 from the expressions you had before. I think this method would be easier than elimination.

 

[SteamKing]

This system originates from systems of linear Diff Eqs. I am required to use the method of undetermined coefficients.

-a2 + 5b2 - a1 = 0
-a2 + b2 -b1 -2 = 0
-a1 + 5b1 +a2 + 1 = 0
-a1 + b1 + b2 = 0

I can't figure out how I am able to solve this. In each case, I have two equations and three unknowns. I know the system is correct.

The answers are:

a2 = -3, b2 = -2/3, a1 = -1/3, b1 = 1/3.

Thanks in advance.

It looks like you have four equations in four unknowns (a1, a2, b1, and b2). Some of the variables just have 0 coefficients in each of the equations. As long as the determinant of the matrix of coefficients for the system is non-zero, you can obtain a unique solution for the system.

 

[enceladus_]

Many thanks, I was able to solve it!

 

[mathman]

This system originates from systems of linear Diff Eqs. I am required to use the method of undetermined coefficients.

-a2 + 5b2 - a1 = 0
-a2 + b2 -b1 -2 = 0
-a1 + 5b1 +a2 + 1 = 0
-a1 + b1 + b2 = 0

I can't figure out how I am able to solve this. In each case, I have two equations and three unknowns. I know the system is correct.

The answers are:

a2 = -3, b2 = -2/3, a1 = -1/3, b1 = 1/3.

Thanks in advance.

I see four equations in four unknowns?

 

[SteamKing]

I see four equations in four unknowns?

You're late to the party. See post #5 above.