Complex roots of the characteristic equations 1 | Second order differential equations | Khan Academy

TL;DR

If a linear differential equation with constant coefficients has complex roots, the general solution involves complex numbers and exponential functions.

Transcript

We learned in the last several videos, that if I had a linear differential equation with constant coefficients in a homogeneous one, that had the form A times the second derivative plus B times the first derivative plus C times-- you could say the function, or the 0 derivative-- equal to 0. If that's our differential equation that the characteristi... Read More

Key Insights

• 🥺 Linear differential equations with constant coefficients can have either real or complex roots, leading to different forms of the general solution.
• 👨‍💼 Complex roots in differential equations result in solutions involving complex numbers and combinations of exponential, cosine, and sine functions.
• 😑 Euler's formula, e^(ix) = cos(x) + isin(x), is used to simplify the general solution and express exponential functions in terms of cosine and sine functions.

Questions & Answers

Q: What is the characteristic equation of a linear differential equation with constant coefficients?

The characteristic equation of a linear differential equation with constant coefficients is obtained by setting the coefficients and the differential equation equal to zero. It is represented as Ar^2 + Br + C = 0.

Q: What are the roots of a characteristic equation with real coefficients?

If the roots of the characteristic equation are real, let's say r1 and r2, then the general solution of the differential equation is y = c1e^(r1x) + c2e^(r2x), where c1 and c2 are constants.

Q: How are complex roots different from real roots in a characteristic equation?

Complex roots occur when the discriminant of the characteristic equation (B^2 - 4AC) is negative. In this case, the roots will have a real part (lambda) and an imaginary part (mu times i).

Q: What is the general solution for differential equations with complex roots?

The general solution for differential equations with complex roots is y = e^(lambda x)(c1cos(mu x) + c2sin(mu x)). It involves exponential functions and a combination of cosine and sine functions due to Euler's formula.

Summary & Key Takeaways

• Linear differential equations with constant coefficients can have complex roots, which result in the general solution involving complex numbers.

• The general solution for differential equations with complex roots includes exponential functions multiplied by a constant and the real and imaginary parts of the complex roots.

• Euler's formula is used to simplify the general solution, expressing exponential functions in terms of cosine and sine functions.

• The general solution for differential equations with complex roots is expressed as a combination of exponential functions, cosine functions, and sine functions.