Intereting Posts

Proving that $n \choose k$ is an integer
Is $\sum\frac1{p^{1+ 1/p}}$ divergent?
proof of l'Hôpital's rule
Boundaries in heat equation
Learning Complex Geometry – Textbook Recommendation Request
Lagrange diagonalization theorem – what if we omit assumption about the form being symmetric
Is there a multiple function composition operator?
Closed-form of integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $
Computing irrational numbers
Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$
Finding P value
Summing over General Functions of Primes and an Application to Prime $\zeta$ Function
Translations of Kolmogorov Student Olympiads in Probability Theory
Are all the norms of $L^p$ space equivalent?
Bounded operator that does not attain its norm

I have calculated solutions to homogenous equations but is the complementary solution mentioned here the same as the homogenous solution?

Let’s take example $y”-3y’+2y=\cos(wx)$ and now the homogenous solution is

$$y_{hom}=C_{1}e^{2x}+C_{2}e^{x}_{|\text{Characteristic eq. =}(r-1)(r-2)}$$

which can be showed with Wronk’s determinant to be valid (cannot yet understand it but go on). Now to find out the general solution there are multiple ways apparently:

- Method of Undetermined Coefficient
- Variation of constant

I have not practised them yet enough (because cannot understand the terms yet) so cannot ask much about them but I am trying to, could someone help me with the terminology here about the `complementary solution`

?

- Generously Feasible?
- Fundamental Theorem of Trigonometry
- In simple English, what does it mean to be transcendental?
- How do you read the symbol “$\in$”?
- What is a real number (also rational, decimal, integer, natural, cardinal, ordinal…)?
- What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?
- Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.
- How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.
- What is the difference between the terms “classical solutions” and “smooth solutions” in the PDE theory?
- Is it possible to write the curl in terms of the infinitesimal rotation tensor?

As Andre mentioned in his comment, the more common terminology is “particular solution”.

Your homogeneous solution $y_{hom} = C_1e^{2x} + C_2e^x$ a solution, not to the original equation, but to the homogeneous equation $y'' – 3y' + 2y = 0$, regardless of the constant parameters $C_1$ and $C_2$.

To find the particular (complementary) solution, we must consider solutions of the form $$y_p=A\cos(wx) + B\sin(wx)\tag{1}$$ After finding $$y_p'=wB\cos(wx) – wA\sin(wx)$$ and $$y_p''=-w^2A\cos(wx) – w^2B\sin(wx)$$ we substitute them into the original equation to get:

$-w^2A\cos(wx) – w^2Bsin(wx) – 3wB \cos(wx) + 3wA\sin(wx) + 2A\cos(wx) + 2Bs\in(wx) = \cos(wx)$

Because $\sin(wx)$ and $\cos(wx)$ are linearly independent, we know that the coefficients of $\cos(wx)$ on the LHS must equal the coefficient of $\cos(wx)$ on the RHS, and similarly for $\sin(wx)$ which gives us the system of equations:

$$\begin {cases} (-w^2 + 2)A + (-3w)B = 1 \\ (3w)A + (-w^2 + 2)B = 0 \end{cases}$$

Solving this system of equations gives us specific values for the coefficients $A$ and $B$:

$$A = \frac{2-w^2}{w^4+5w^2+4}, B = \frac{-3w}{w^4+5 w^2+4}$$

(provided that the denominator of those fractions is non-zero of course – also notice that that is equivalent to the condition that the determinant of the coefficient matrix for $A$ and $B$, in the system of equations given above, is non-zero.)

which now provides us with a particular (complimentary) solution to the original differential equation:

$$y_p=\frac{2-w^2}{w^4+5w^2+4}cos(wx) + \frac{-3w}{w^4+5 w^2+4}sin(wx)$$

And because, in general, (f + g)’ = f’ + g’, we can see that

$$ (y_{hom} + y_p)'' – 3(y_{hom} + y_p)' + 2(y_{hom} + y_p) = (y_{hom}'' – 3y_{hom}' + 2y_{hom}) + (y_p'' -3y_p' +2y_p)$$

and then

$$(y_{hom}'' – 3y_{hom}' + 2y_{hom}) + (y_p'' -3y_p' +2y_p) = 0 + cos(wx) = cos(wx)$$

thus the general solution will be

$$y=y_{hom}+y_p=C_1e^{2x} + C_2e^x + \frac{2-w^2}{w^4+5w^2+4}\cos(wx) + \frac{-3w}{w^4+5 w^2+4}\sin(wx)$$

I just wanted to answer the original question and point something out. In every class I’ve ever had, the complementary solution is the solution to the “homogeneous equation” and is often (if incorrectly) referred to as the homogeneous solution. So in this case I believe you are correct. The original equation is not itself homogeneous; however, the “homogeneous solution” is the solution for the original equation without a driving function (RHS = 0). And you’ve put that as your answer.

**Another point, as homogeneous solution and complementary solution are often used interchangeably, “particular solution” and “complimentary solution” are definitely *not* used interchangeably. In several answers and comments, people sound is if they refer to the same thing when they do not. For any linear ordinary differential equation, the general solution (for all t for the original equation) can be represented as the sum of the complementary solution and the particular solution.

Vg(t)=Vp(t)+Vc(t)

In electrical engineering context, the complementary and particular solutions have their own names (because we always rename things), the particular solution is usually called the “steady-state response” and the complementary solution is called the “transient response”.

- Is this a perfect set?
- How to find an angle in range(0, 360) between 2 vectors?
- Product of sets and supremum
- Is every non-square integer a primitive root modulo some odd prime?
- Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.
- Find control point on piecewise quadratic Bézier curve
- Inverse of an invertible triangular matrix (either upper or lower) is triangular of the same kind
- Topology on $R((t))$, why is it always the same?
- How to show that for any abelian group $G$, $\text{Hom}(\mathbb{Z},G)$ is isomorphic to $G$
- Analytic floor function, why this seems to work?
- Lie algebra and left-invariant vector fields
- Why is determinant a multilinear function?
- Geometric meaning of primary decomposition
- Littlewood's Inequality
- Distance to a closed set is continuous.