2nd order homogeneous equations complex root

In summary, a 2nd order homogeneous equation having complex roots means that the solutions involve imaginary numbers and cannot be expressed as real numbers. Complex roots are typically represented as a+bi, where a and b are real numbers and i is the imaginary unit. Knowing if a 2nd order homogeneous equation has complex roots is important for understanding the behavior of the solutions and gaining insights into the represented systems. To solve such an equation, the quadratic formula or the method of undetermined coefficients can be used, with the complex roots expressed as a+bi. It is also possible for a 2nd order homogeneous equation to have both real and complex roots when the discriminant is 0, resulting in a double root that is both real and complex
  • #1
Larrytsai
228
0

Homework Statement


y'' -2y' +5y =0 , y(0)=1, y'(0)=1
you get a complex root conjugate.

Homework Equations


y=e^(rt)
y'=re^(rt)
y''=r^2 * e^(rt)

The Attempt at a Solution


I have in my notes sin(omega*t)e^(sigma *t), cos(omega *t)e^(sigma).
I don't think i took down the notes properly, can anyone care confirm if this is right or wrong?
 
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  • #2
Yes your homogeneous solutions are sin(ωt)eσt and cos(ωt)eσt when the roots are in the form r=σ ± ωi with 'i' being the imaginary unit.
 

FAQ: 2nd order homogeneous equations complex root

What does it mean for a 2nd order homogeneous equation to have complex roots?

When a 2nd order homogeneous equation has complex roots, it means that the solutions to the equation involve imaginary numbers. In other words, the solutions cannot be expressed as real numbers.

How are complex roots of 2nd order homogeneous equations represented?

Complex roots are typically represented in the form of a+bi, where a and b are real numbers and i is the imaginary unit (i.e. the square root of -1).

What is the importance of knowing if a 2nd order homogeneous equation has complex roots?

Knowing if a 2nd order homogeneous equation has complex roots can help in understanding the behavior of the solutions to the equation. It can also provide insights into the physical or mathematical systems that the equation represents.

How do you solve a 2nd order homogeneous equation with complex roots?

To solve a 2nd order homogeneous equation with complex roots, you can use the quadratic formula or the method of undetermined coefficients. In both methods, the complex roots will be expressed in the form of a+bi.

Can a 2nd order homogeneous equation have both real and complex roots?

Yes, it is possible for a 2nd order homogeneous equation to have both real and complex roots. This happens when the discriminant of the equation is equal to 0, which results in a double root that is both real and complex.

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