Inhomogenous 2nd Order Linear Differentials

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Table of Contents

• 4.3.1: Inhomogeneous with Constant Coefficients
  • Key Definitions
    • $y^{″}+p{y}^{\prime }+qy=g(t)$
  • General Solution for Constant Coefficients
    • $y(t;C_1,C_2)=C_1{y}_1(t)+C_2{y}_2(t)+y_p(t)$

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4.3.1: Inhomogeneous with Constant Coefficients

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Key Definitions

Inhomogeneous Equations - Eq. w/ forcing term $g(t)\ne 0$ I.e. dealing with when $\ne 0$ of form:

$y^{″}+p{y}^{\prime }+qy=g(t)$

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General Solution for Constant Coefficients

If $y_p$ is a particular solution to the inhomogeneous eq. $y^{\prime }‘+p{y}^{\prime }+qy=g(t)$ and $y_1,y_2$ form a fundamental set of solutions to the homogeneous eq. $y^{\prime }‘+p{y}^{\prime }+qy=0$, then the general solution is:

$y(t;C_1,C_2)=C_1{y}_1(t)+C_2{y}_2(t)+y_p(t)$

Use to find a particular solution if $p,q$ are constant Use otherwise