Solving the Differential Equation
Problem: Solve the differential equation
Explanation:
The equation is a second-order linear homogeneous differential equation with constant coefficients. Its general form is
First, we assume a solution of the form
Upon differentiating, we find:
- First derivative:
- Second derivative:
Substituting into the original equation, we get
Since
Therefore, the general solution becomes
About Linear Independence:
The functions
Solving the Differential Equation
Problem: Solve the differential equation
Explanation:
The equation is a second-order linear homogeneous differential equation with constant coefficients. Its general form is
First, we assume a solution of the form
Upon differentiating, we find:
- First derivative:
- Second derivative:
Substituting into the original equation, we get
Since
Therefore, the general solution becomes