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We now consider general nonminushomogeneous equation,

Â nonminushomogeneous differential equation one.

Â Let's remind that the differential equation one here.

Â Differential equation one is L_y which is,

Â let me use this summation notation.

Â Sum of i is equal to zero to the n,

Â and a_i_x and D_to_the_i_of_y and that is equal to b_x.

Â This is the differential equation.

Â The nonminushomogeneous differential equation one.

Â For this one, first now I'd like to

Â introduce to you the sominuscalled superposition principal,

Â superposition principal for nonminushomogeneous differential equation.

Â So, I'm considering this where in particular I'm concerning

Â with the differential operator l which is the sum of

Â the differential polynomial say a_i times D_to_the_i.

Â Let's assume

Â that L_phi_i equals b_i_x.

Â On some interval i,

Â for i from one to k,

Â where the L is the differential polynomial in two.

Â Then my claim is that L of the linear combinations sum of c_i times phi_i.

Â There is a sum of c_i times b_i for arbitrary constant c_i.

Â The easy proof I just give.

Â It's because of the linearity

Â of this differential operator.

Â So it's almost trivial thing.

Â Now, it's time to consider the general solution of

Â nonminushomogeneous linear differential equation.

Â Nonminushomogeneous linear differential equation exactly the differential equation one.

Â So far we considered only when b_x equals zero,

Â the sominuscalled homogeneous linear equation.

Â Now, we are considering the nonminushomogeneous differential equation.

Â Let y_p, p is coming from the particular.

Â y_p be any particular solution of the nonminushomogeneous equation one on the interval i.

Â Any one particular solution,

Â than the claim is that the general solution of

Â this nonminushomogeneous equation can

Â be written as in symbol y equals y_c plus y_p, what is y_c?

Â y_c is the sum of c_y times y_i,

Â where the c_y are the arbitrary constants and

Â the where the y_i is any fundamental set of

Â solutions of the corresponding homogeneous differential equation four.

Â Say L_y equals zero on i.

Â Then you can recognize

Â immediately down there this summation, the first part.

Â This is a general solution of corresponding homogeneous problem.

Â Because y_i they form a fundamental set of solutions for this homogeneous problem.

Â This summation I denoted by y_c because the people call this part

Â to be a complimentary solution of the differential equation one.

Â Here's a very easy argument, easy proof.

Â Now for arbitrary leader land.

Â Now, let y be any solution of the given differential equation,

Â L_y equals b, L_y equals b,

Â with which compute L_y

Â minus y(p) because L is a linear operator.

Â It's the same as the L_y minus L_y_p.

Â How much is L_y because y is a solution.

Â L_y is the b_x.

Â How much is the L_y_p.

Â Because y_p is a particular solution of the same equation.

Â So L_y_p is also b_x.

Â Their subtraction must be identically 0. What does that mean?

Â y minus y_p must be a solution of

Â corresponding homogeneous problem, corresponding homogeneous problem.

Â Then it means, this y minus y_p

Â must be a linear combination of members of the fundamental set.

Â That is why it c_i.

Â So you get this linear combination for suitable constant c_i.

Â And then this means,

Â the arbitrary solution of differential equation one,

Â which is i denoted by y,

Â y must be y_p plus sum of c_i times y_i.

Â That is the conclusion.

Â That's the end of the proof.

Â Moreover, we can see that the family of solutions,

Â the family of solutions in equation seven.

Â What is the equation seven? Right here.

Â This family of solution. In fact, right here.

Â This is the set of

Â all possible solutions of nonminushomogeneous differential equation one.

Â As I said before,

Â we say, we call the y_c,

Â which is the combination of the members

Â of the fundamental set of solutions for corresponding homogeneous problem.

Â We call this y_c as

Â the complementary solution of nonminushomogeneous differential equation.

Â So I'm denoting it,

Â people denoted by y_c.

Â