Second-order nonhomogeneous differential equations initial value problems

To solve an initial value problem for a second-order nonhomogeneous differential equation, we’ll follow a very specific set of steps. We first find the complementary solution, then the particular solution, putting them together to find the general solution. Then we differentiate the general solution, plug the given initial conditions into the general solution and its derivative to create a system of linear equations, and then use the initial conditions to solve that system for the constant coefficients. Finally, we’ll plug those constant coefficients back into the general solution.

How to solve initial value problems

We always lose the constant (term without a variable attached), when we take the derivative of a function. Which means we’re never going to get the constant back when we try to integrate our derivative. It’s lost forever. That is, unless we have an initial condition we can use to figure out what that constant was before we differentiated.

Solving initial value problems using laplace transforms

To use a Laplace transform to solve a second-order nonhomogeneous differential equations initial value problem, we’ll need to use a table of Laplace transforms or the definition of the Laplace transform to put the differential equation in terms of Y(s). Once we solve the resulting equation for Y(s), we’ll want to simplify it until we recognize that the terms in our equation match formulas in a table of Laplace transforms.

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