Solving 4th Order Diff. Eq. with Complex Root: Daunting Task?

In summary, the problem is to solve a 4th order differential equation with a given complex root. To find the homogenous solution, one must consider the coefficients of the equation and use the knowledge of solving quadratic equations with complex roots. The two given roots are -2+3i and -2-3i, and by creating a quadratic equation with these roots, we can find the other two roots. Ultimately, this will lead to finding the homogenous solution.
  • #1
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I have a 4th order differential equation with given -2 +3i root.

Now need to find the homogenous solution. Well, if the root was real, it would be easier but now I'm stuck and don't know how to proceed.

What am I supposed to do to solve this ?

Equation is : d4y(t)/dt4 +6d3y(t)/dt3 + 22d2y(t)/dt2 + 30dy(t)/dt + 13y(t) = f(t)

Just need to solve the homogenous part so f(t) is just a dummy function
 
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  • #2
The coefficients are real, so actually you have been given two complex roots, not one.

If you haven't done any formal courses about roots of polynomials, think about what you get when you solve a quadratic equation with a pair of complex roots. That should lead you to finding a quadratic factor (with real coefficients) of the 4th-order equation.
 
  • #3
AlephZero said:
The coefficients are real, so actually you have been given two complex roots, not one.

If you haven't done any formal courses about roots of polynomials, think about what you get when you solve a quadratic equation with a pair of complex roots. That should lead you to finding a quadratic factor (with real coefficients) of the 4th-order equation.

You're right there are two roots are given -2 - 3i and -2 + 3i
I tried to recreate a quadratic equation with those roots by assuming coefficient of a=1, b=4 and c=13. However it didn't work.
 
  • #4
Ok, I've just done it, thanks though.
 
  • #5
Great! For those who are interested, though, let me note that since [itex]-2+3i[/itex] and [itex]-2-3i[/itex] are roots then [itex](x- (-2+3i))(x-(-2-3i))= (x+2- 3i)(x+2+ 3i)= (x+2)^2- (3i)^2= x^2+ 4x+ 4+ 9= x^2+ 4x+ 13[/itex]. Now divide [itex]x^4+ 6x^3+ 22x^2+ 30x+ 13[/itex] by that to find the quadratic equation the other roots must satisfy.
 

FAQ: Solving 4th Order Diff. Eq. with Complex Root: Daunting Task?

1. What is a 4th order differential equation?

A 4th order differential equation is an equation that involves the fourth derivative of a function. It is a mathematical equation that describes the relationship between a function and its derivatives.

2. What makes solving a 4th order differential equation with complex roots challenging?

The presence of complex roots in a 4th order differential equation makes it challenging because complex roots involve imaginary numbers, which can be difficult to work with and understand. Additionally, the higher order of the equation adds complexity to the solution process.

3. What are some common techniques for solving 4th order differential equations with complex roots?

Some common techniques for solving 4th order differential equations with complex roots include using the characteristic equation, substitution method, and using the Laplace transform. It is also helpful to have a strong understanding of complex numbers and their properties.

4. How does solving 4th order differential equations with complex roots relate to real-world applications?

4th order differential equations with complex roots are often used to model real-world phenomena, such as oscillating systems, electrical circuits, and fluid dynamics. Solving these equations allows us to understand and predict the behavior of these systems.

5. What are some tips for successfully solving 4th order differential equations with complex roots?

Some tips for solving 4th order differential equations with complex roots include breaking the equation down into smaller, more manageable parts, using appropriate techniques for solving complex numbers, and practicing regularly to improve understanding and problem-solving skills.

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