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We now consider general nonminushomogeneous equation,

nonminushomogeneous differential equation one.

Let's remind that the differential equation one here.

Differential equation one is L_y which is,

let me use this summation notation.

Sum of i is equal to zero to the n,

and a_i_x and D_to_the_i_of_y and that is equal to b_x.

This is the differential equation.

The nonminushomogeneous differential equation one.

For this one, first now I'd like to

introduce to you the sominuscalled superposition principal,

superposition principal for nonminushomogeneous differential equation.

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

with the differential operator l which is the sum of

the differential polynomial say a_i times D_to_the_i.

Let's assume

that L_phi_i equals b_i_x.

On some interval i,

for i from one to k,

where the L is the differential polynomial in two.

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

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

The easy proof I just give.

It's because of the linearity

of this differential operator.

So it's almost trivial thing.

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

nonminushomogeneous linear differential equation.

Nonminushomogeneous linear differential equation exactly the differential equation one.

So far we considered only when b_x equals zero,

the sominuscalled homogeneous linear equation.

Now, we are considering the nonminushomogeneous differential equation.

Let y_p, p is coming from the particular.

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

Any one particular solution,

than the claim is that the general solution of

this nonminushomogeneous equation can

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

y_c is the sum of c_y times y_i,

where the c_y are the arbitrary constants and

the where the y_i is any fundamental set of

solutions of the corresponding homogeneous differential equation four.

Say L_y equals zero on i.

Then you can recognize

immediately down there this summation, the first part.

This is a general solution of corresponding homogeneous problem.

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

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

to be a complimentary solution of the differential equation one.

Here's a very easy argument, easy proof.

Now for arbitrary leader land.

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

L_y equals b, L_y equals b,

with which compute L_y

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

It's the same as the L_y minus L_y_p.

How much is L_y because y is a solution.

L_y is the b_x.

How much is the L_y_p.

Because y_p is a particular solution of the same equation.

So L_y_p is also b_x.

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

y minus y_p must be a solution of

corresponding homogeneous problem, corresponding homogeneous problem.

Then it means, this y minus y_p

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

That is why it c_i.

So you get this linear combination for suitable constant c_i.

And then this means,

the arbitrary solution of differential equation one,

which is i denoted by y,

y must be y_p plus sum of c_i times y_i.

That is the conclusion.

That's the end of the proof.

Moreover, we can see that the family of solutions,

the family of solutions in equation seven.

What is the equation seven? Right here.

This family of solution. In fact, right here.

This is the set of

all possible solutions of nonminushomogeneous differential equation one.

As I said before,

we say, we call the y_c,

which is the combination of the members

of the fundamental set of solutions for corresponding homogeneous problem.

We call this y_c as

the complementary solution of nonminushomogeneous differential equation.

So I'm denoting it,

people denoted by y_c.