Question

Solve the differential equation. $25 y^{\prime \prime}+9 y=0$

   Solve the differential equation.
$25 y^{\prime \prime}+9 y=0$

Calculus

James Stewart 7th Edition

Chapter 17, Problem 6

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Solved by Expert Linh Vu

10/11/2020

Step 1

The given equation is a second-order linear homogeneous differential equation with constant coefficients: \[ 25 y^{\prime \prime} + 9 y = 0 \] Show more…

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Solve the differential equation. $25 y^{\prime \prime}+9 y=0$

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Key Concepts

Linear Homogeneous Differential Equation

A linear homogeneous differential equation is one in which the dependent variable and its derivatives appear linearly and there is no forcing term (i.e., the equation equals zero). This property allows for the principle of superposition to apply, meaning that any linear combination of solutions is also a solution.

Constant Coefficients

In differential equations with constant coefficients, the multipliers for the variable and its derivatives do not change with the independent variable. This uniformity simplifies the analysis and solution of the equation, as it enables the use of systematic methods like the characteristic equation.

Characteristic Equation

The characteristic equation is derived by assuming a solution of the form e^(rt) for the differential equation, which converts the differential equation into an algebraic equation in terms of r. Solving this algebraic equation provides the roots that determine the structure of the general solution, whether in real or complex form.

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