Question

Solve the following differential equations. . $(D^2-2D+1)y=0$

Short answer

The solution is $y(t)=(C_1+C_{2}t)e^t$

Step-by-step solution

- Identify the characteristic equation

Rewrite the differential equation in terms of the characteristic equation. For the differential equation $(D^2-2D+1)y=0$, the equivalent characteristic equation is $r^2-2r+1=0$.

- Solve characteristic equation

Solve the characteristic equation $r^2-2r+1=0$. This is a quadratic equation which can be factored as $(r-1{)}^2=0$. Therefore, the roots are $r=1$ with multiplicity 2.

- Write the general solution

For a characteristic equation with a repeated root (in this case, $r=1$ repeated), the general solution to the differential equation is given by $y(t)=(C_1+C_{2}t)e^{rt}y(t)=(C_1+C_{2}t)e^{rt}$. Since $r=1$, the final general solution is $$y(t)=(C_1+C_{2}t)e^t$$

Key concepts

characteristic equation

The characteristic equation is a fundamental concept when solving linear differential equations. It connects the differential equation to a polynomial, making it easier to find solutions. In our example, we start with the second-order differential equation: $$(D^2-2D+1)y=0$$By substituting the operator $D$ with $r$, we can rewrite it as a polynomial, called the characteristic equation:$$r^2-2r+1=0$$This transformation allows us to solve for $r$ using algebraic methods, such as factoring or the quadratic formula. Finding the roots $r$ is the key to determining the general solution of the differential equation.

general solution

The general solution of a second-order differential equation depends on the roots of its characteristic equation. After solving the characteristic equation, we found the roots: $$r=1$$Since this is a repeated root (multiplicity 2), we use a specific form to represent the general solution. For repeated roots, the general solution is given by: $$y(t)=(C_1+C_{2}t)e^{rt}$$Here, $C_1$ and $C_2$ are constants determined by initial conditions. In our case, with $r=1$, the solution becomes: $$y(t)=(C_1+C_{2}t)e^t$$This formula encompasses all possible solutions to the original differential equation.

repeated roots

Repeated roots occur when solving the characteristic equation yields the same root more than once. In our example, the characteristic equation $$r^2-2r+1=0$$factors as $$(r-1{)}^2=0$$indicating a repeated root at $r=1$. Repeated roots require a special treatment in forming the general solution. Instead of simply using exponential functions, we introduce an extra term involving the variable $t$, leading to the form: $$y(t)=(C_1+C_{2}t)e^{rt}$$This ensures the solution captures all behaviors represented by the differential equation. The additional $t$ term is essential in accounting for the root's multiplicity.

second-order differential equation

Second-order differential equations involve derivatives up to the second order. In general, they can be written as: $$a{y}^{″}+b{y}^{\prime }+cy=0$$where $a$, $b$, and $c$ are constants. The specific example we solved is: $$(D^2-2D+1)y=0$$Here, $D$ represents differentiation with respect to $t$. Solving these equations typically involves:

• Formulating the characteristic equation
• Finding its roots
• Using the roots to write the general solution

Second-order differential equations show up in many physical phenomena such as oscillations, circuits, and mechanical systems, making them essential in various scientific and engineering fields.