Question

Prove: If $$y=c_1{y}_1+c_2{y}_2+\cdots +c_k{y}_k+y_p$$ is a solution of a linear equation $Ly=F$ for every choice of the constants $c_1,c_2,\dots ,c_k,$ then $L{y}_i=0$ for $1\le i\le k$

Short answer

Question: Prove that if a function y which is a linear combination of k+1 functions yi satisfies a linear equation Ly = F for any choice of coefficients ci, then L yi = 0 for 1 ≤ i ≤ k. Solution: We showed that the given condition leads to L yi = 0 for 1 ≤ i ≤ k by substituting the expression for y into the linear equation Ly = F, applying the linearity property of linear operators, and then choosing specific constants ci. Since L(yj) = 0 for any arbitrary index in the range 1 ≤ j ≤ k, this implies that L yi = 0 for 1 ≤ i ≤ k, as required.

Step-by-step solution

Substitute the expression for $y$ into the linear equation

First, let's substitute the expression for $y$ into the given linear equation $Ly=F$. We have: $$L(c_1{y}_1+c_2{y}_2+\cdots +c_k{y}_k+y_p)=F$$

Apply the Linearity Property of Linear Operators

Since $L$ is a linear operator, it satisfies the linearity property, which means that for any constants $c_i$ and functions $y_i$, it holds that: $$L(c_1{y}_1+c_2{y}_2+\cdots +c_k{y}_k+y_p)=c_{1}L(y_1)+c_{2}L(y_2)+\cdots +c_{k}L(y_k)+L(y_p)$$ Now, the given condition states that the above equation equals $F$ for any choice of constants $c_1,c_2,\dots ,c_k$. That means: $$c_{1}L(y_1)+c_{2}L(y_2)+\cdots +c_{k}L(y_k)+L(y_p)=F$$

Show that $L{y}_i=0$ for $1\le i\le k$

Let's consider the following choice of constants, where for some $1\le j\le k$, we have $c_j=0$ and $c_i=1$ for all $i\ne j$. Then, the above equation becomes: $$c_{1}L(y_1)+\cdots +0\cdot L(y_j)+\cdots +c_{k}L(y_k)+L(y_p)=F$$ Removing the zero term we get: $$c_{1}L(y_1)+\cdots +c_{j-1}L(y_{j-1})+c_{j+1}L(y_{j+1})+\cdots +c_{k}L(y_k)+L(y_p)=F$$ Now, we subtract this equation from the original equation, and note that the term $L(y_p)$ cancels out: $$c_{j}L(y_j)=0$$ Since this equation holds for any choice of constants $c_i$, and for particular $c_j=1$, we have: $$L(y_j)=0$$ Since $j$ is an arbitrary index in the range $1\le j\le k$, this implies that $L{y}_i=0$ for $1\le i\le k$. This completes the proof.

Key concepts

Homogeneous Solutions

When we deal with linear differential equations, a special type of solution we often encounter is called the 'homogeneous solution'. In a nutshell, a homogeneous solution to a differential equation is one where the function on the right-hand side is zero, essentially representing a system with no external forces or sources. This kind of solution is critical because it reflects the intrinsic behavior of the system.

In our exercise, the term 'homogeneous solution' refers to the functions $y_i$ such that $L(y_i)=0$. These solutions make up the complementary part of the general solution to the differential equation. For a linear differential equation, the homogeneous solutions can be combined through multiplication by constants and addition to construct more complex solutions, a concept stemming from the superposition principle, which we will explore in depth later on.

Linearity Property of Linear Operators

The linearity property of linear operators is fundamental in understanding how linear differential equations work. A linear operator, denoted usually by $L$, behaves predictably when it interacts with functions and constants through addition and scalar multiplication.

In the context of our exercise, $L$ being a linear operator indicates that we can distribute it across a sum of functions and pull out constants from within its operation. Mathematically, this behavior is expressed as $L(c_1{y}_1+c_2{y}_2+\cdots +c_k{y}_k+y_p)=c_{1}L(y_1)+c_{2}L(y_2)+\cdots +c_{k}L(y_k)+L(y_p)$, and lays the groundwork for proving that the functions $y_i$ are indeed homogeneous solutions. $L$'s linearity is essential since it allows us to manipulate and simplify complex expressions within the differential equation.

Superposition Principle

The superposition principle is a diamond in the rough for students working through linear differential equations. This principle says that if we have a set of solutions to a linear differential equation, we can create new solutions by taking any linear combination of these solutions, i.e., by adding them together after multiplying each by a constant. This makes it an invaluable tool for constructing the general solution of a linear differential equation.

In our example problem, by applying the superposition principle, we were able to show that the linear combination $y=c_1{y}_1+c_2{y}_2+\cdots +c_k{y}_k+y_p$ holds as a solution for any constants $c_1,c_2,\dots ,c_k$. This principle underscores the value of homogeneous solutions and demonstrates the inherent linearity of the problem at hand.

Differential Equations Proof

Proving properties about solutions to differential equations often involves a mixture of algebraic manipulation and applying concepts specific to differential equations, such as the superposition principle and linearity property of linear operators.

In our step-by-step solution, we used these ideas to prove that individual functions $y_i$ must be homogeneous solutions if the linear combination $y$ is a solution for any choice of constants. By strategically choosing values for the constants $c_i$ and leveraging the linear operator's behavior, we showed that each $y_i$ satisfies the homogeneous equation $L(y_i)=0$. This proof not only reinforces the understanding of linearity but also illustrates the methodical approach needed to prove assertions in the context of differential equations. Through such proofs, students learn how the theoretical underpinnings of differential equations are reflected in the equations' solutions.