Solve the following homogeneous linear ordinary differential...

Solve the following homogeneous linear ordinary differential equations. What is the dimension of the solution space? (a) $u^{\prime \prime}-4 u=0$, (b) $u^{\prime \prime}-6 u^{\prime}+8 u=0$, (c) $u^{\prime \prime \prime}-9 u^{\prime}=0$, (d) $u^{\prime \prime \prime \prime}+4 u^{\prime \prime \prime}-u^{\prime \prime}-16 u^{\prime}-12 u=0$.

Solve the following homogeneous linear ordinary differential equations. What is the dimension of the solution space? (a) $u^{\prime \prime}-4 u=0$, (b) $u^{\prime \prime}-6 u^{\prime}+8 u=0$,
(c) $u^{\prime \prime \prime}-9 u^{\prime}=0$,
(d) $u^{\prime \prime \prime \prime}+4 u^{\prime \prime \prime}-u^{\prime \prime}-16 u^{\prime}-12 u=0$.

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Key Concepts

Homogeneous Linear Ordinary Differential Equations

These are differential equations in which the dependent variable and all its derivatives appear linearly and there is no nonhomogeneous (forcing) term. Their structure allows the use of the principle of superposition, meaning that any linear combination of solutions is also a solution.

Characteristic Equation

This is obtained by substituting a trial solution (usually an exponential function) into the differential equation, converting it into an algebraic equation. The roots of this characteristic equation determine the form of the general solution, where each distinct root leads to an independent solution, and repeated roots require additional terms to achieve independence.

Order of the Differential Equation and Dimension of the Solution Space

The order of a linear differential equation (i.e., the highest derivative present) directly corresponds to the number of independent solutions required to form the general solution. Consequently, the solution space is a vector space whose dimension is equal to the order of the differential equation, ensuring that the complete set of linearly independent solutions spans the space.

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