Higher Order Differential Equation when R.H.S = 0 - Problem 2 | Summary and Q&A

TL;DR

Learn how to find the roots and solution for higher order differential equations with constant coefficients and a zero right hand side.

Key Insights

• ✋ The steps for solving higher order differential equations with zero right hand side involve finding the complementary function (yc) and the particular integral (yp).
• 🫚 Synthetic division can be used to find the roots of the auxiliary equation in a degree 4 equation.
• 👶 The complementary function (yc) accounts for the homogeneous solutions of the differential equation.
• 🫱 The particular integral (yp) is zero when the right hand side is zero, resulting in the final solution being the complementary function only.

Transcript

Read and summarize the transcript of this video on Glasp Reader (beta).

Questions & Answers

Q: Why do we start by finding the complementary function (yc) in higher order differential equations?

We find the complementary function to account for the homogeneous solutions of the differential equation. It is found by solving the auxiliary equation, which helps us find the roots.

Q: What is the particular integral (yp) and why is it zero in this case?

The particular integral represents the particular solution of the differential equation. In this case, since the right hand side is zero, the particular integral is also zero.

Q: How do we find the roots of the auxiliary equation in a higher order differential equation?

For a degree 4 equation like in this example, synthetic division can be used to find the roots. By applying synthetic division successively and considering different roots, all four roots can be obtained.

Q: What is the final solution for the given higher order differential equation?

The final solution is the complementary function (yc) since the particular integral (yp) is zero. It can be expressed as yc = c1e^x + c2e^-x + c3e^2x + c4e^3x.

Summary & Key Takeaways

• The video explains the steps to solve higher order differential equations when the right hand side is zero.

• The first step is to find the complementary function (yc) by finding the roots of the auxiliary equation.

• The particular integral (yp) is zero in this case, so the final solution is yc.