Second-Order Linear ODEs With Constant Coefficients

2 Second-Order Linear ODEs With Constant Coefficients 

The hrsl-order system considered in the previous section yields well-behaved exponential responses. Second-order systems can be much more exciting since they can give an oscillatory or underdamped response. 

The first-order linear equation [Eq. (6.44)] could have a time-variable coefficient that is, 0) could be a function of time. We will consider only linear second-order ODEs that have constant coefficients (tj, and ( are constants). 

Analytical methods are available for linear ODEs with variable coeificients, but their solutions are usually messy infinite series. We will not consider them here. 

The solution of a second-order ODE can be deduced from the solution of a first-order ODE. Equation (6.45) can be broken up into two parts  

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Second-Order Linear ODEs with Constant Coefficients... 

First-Order Linear Ordinary Differential Equation / 2.3.2 Second-Order Linear ODEs with Constant Coefficients / 2.3.3 Nth-Order Linear ODEs with Constant Coefficients... 

Equation 9.18 is a linear, second-order ODE with constant coefficients. An analytical solution is possible when the reaction is first order. The general solution to Equation 9.18 with 3 A = —ka is... 

The mass balance with diffusion and first-order chemical reaction, given by (24-12), is classified as a frequently occurring second-order linear ordinary differential equation (i.e., ODE) with constant coefficients. It is a second-order equation because diffusion is an important mass transfer rate process that is included in the mass balance. It is linear because the kinetic rate law is first-order or pseudo-first-order, and it is ordinary because diffusion is considered only in one coordinate direction—normal to the interface. The coefficients are constant under isothermal conditions because the physicochemical properties of the fluid don t change... 

The ODEs are linear with constant coefficients. They can be converted to a single, second order ODE, much like Equation (11.22), if an analytical solution is desired. A numerical solution is easier and better illustrates what is necessary for anything but the simplest problem. Convert the independent variable to dimensionless form, = z/L. Then... 

Equations (11.20) and (11.21) are linear, first-order ODEs with coefficients that are assumed constant. The equations can be combined to give a second-order ODE in af. 

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