Solve the following inhomogeneous linear ordinary...

Solve the following inhomogeneous linear ordinary differential equations:
(a) $u^{\prime}-4 u=x-3$,
(b) $5 u^{\prime \prime}-4 u^{\prime}+4 u=e^x \cos x$,
(c) $u^{\prime \prime}-3 u^{\prime}=e^{3 x}$.

Key Concepts

Linear Ordinary Differential Equations

This concept involves equations that relate a function to its derivatives where all terms depend linearly on the unknown function and its derivatives. Understanding linear ODEs is crucial because they possess properties like superposition, which allow solutions to be expressed as sums of simpler, often independent, components.

Inhomogeneity in Differential Equations

An inhomogeneous differential equation includes a non-zero forcing term or source term, which makes the equation non-homogeneous. The solution approach typically involves finding the general solution by combining the homogeneous (complementary) solution with a particular solution that specifically accounts for the inhomogeneous terms.

Integrating Factor Method

The integrating factor method is a technique used primarily for first-order linear ordinary differential equations. It involves multiplying the entire equation by an integrating factor, which simplifies the left-hand side to the derivative of a product, making it straightforward to integrate and solve for the unknown function.

Characteristic Equation and Homogeneous Solutions

When solving linear ODEs with constant coefficients, the homogeneous part of the equation is solved by assuming exponential solutions, leading to a polynomial known as the characteristic equation. The roots of this equation determine the form of the homogeneous solution, whether it involves distinct real exponentials, repeated roots, or complex conjugate pairs.

Method of Undetermined Coefficients

This is a systematic technique for finding a particular solution to an inhomogeneous linear ODE with constant coefficients when the forcing term is of a type whose derivatives reproduce a similar form (like polynomials, exponentials, sines, and cosines). By guessing a form for the particular solution and determining the coefficients through substitution into the original equation, one can effectively solve the ODE.

Superposition Principle

The superposition principle states that the general solution to a linear inhomogeneous differential equation can be written as the sum of the general solution to the corresponding homogeneous equation (complementary solution) and any particular solution of the inhomogeneous equation. This principle underpins many solution strategies for linear ODEs.

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