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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.