Question

Solve the differential equation. $y^{\prime \prime}-4 y^{\prime}+y=0$

   Solve the differential equation.
 $y^{\prime \prime}-4 y^{\prime}+y=0$

Calculus

James Stewart 7th Edition

Chapter 17, Problem 8

Instant Answer

Solved by Expert Raj Bala

01/24/2022

Step 1

Step 1: We have a second-order linear homogeneous differential equation: $y'' - 4y' + y = 0$ To solve this, we need to find the general solution by first determining the characteristic equation. Show more…

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Solve the differential equation. $y^{\prime \prime}-4 y^{\prime}+y=0$

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Key Concepts

Second Order Linear Homogeneous Differential Equations

These are differential equations of the form a y'' + b y' + c y = 0, where a, b, and c are constants. Their solutions typically involve finding two linearly independent solutions, which can then be combined to form the general solution, reflecting the principle of superposition.

Characteristic Equation

This is a polynomial equation obtained by assuming a solution of the exponential form (e.g., y = e^(rx)). By substituting this form into the differential equation, one derives a polynomial equation in r. The roots of this characteristic equation determine the behavior of the solution, indicating whether the solution consists of real exponentials, complex exponentials, or repeated roots.

General Solution Formation

Once the roots of the characteristic equation are determined, the general solution is constructed as a linear combination of the independent solutions corresponding to those roots. When the roots are distinct, the general solution is a sum of exponential functions multiplied by arbitrary constants, encapsulating all possible solutions to the differential equation.

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