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When the characteristic roots $r_1$ and $r_2$ are real and distinct, each root generates a valid exponential solution $e^{r_1 t}$ and $e^{r_2 t}$. Because the differential equation is linear and homogeneous, the superposition principle applies, meaning any linear combination of these solutions is also a solution.

Conditions: The differential equation is linear, homogeneous, and has constant coefficients.; The characteristic roots $r_1$ and $r_2$ are real and distinct.

When the characteristic roots $r_1$ and $r_2$ are real and distinct, each root generates a valid exponential solution $e^{r_1 t}$ and $e^{r_2 t}$. Because the differential equation is linear and homogeneous, the superposition principle applies, meaning any linear combination of these solutions is also a solution.

Conditions: The differential equation is linear, homogeneous, and has constant coefficients.; The characteristic roots $r_1$ and $r_2$ are real and distinct.

Euler's formula is used to convert the exponential solutions with complex exponents into real-valued trigonometric functions (sines and cosines). This transformation makes the solution physically interpretable and mathematically manageable for real-world applications where the output $y(t)$ must be real.

Conditions: The characteristic roots are complex.; The original differential equation has real coefficients.

Euler's formula is used to convert the exponential solutions with complex exponents into real-valued trigonometric functions (sines and cosines). This transformation makes the solution physically interpretable and mathematically manageable for real-world applications where the output $y(t)$ must be real.

Conditions: The characteristic roots are complex.; The original differential equation has real coefficients.