Lesson: Linear Homogeneous Differential Equations with Constant Coefficients

Mathematics

In this lesson, we will learn how to solve second- and higher-order linear homogeneous differential equations with constant coefficients.

Worksheet

Q1:

Suppose there are differentiable functions 𝐢 and 𝑆, defined for all real numbers and satisfying 𝐢=π‘†οŽ˜ and 𝑆=𝐢.

From the derivatives, what can you deduce about πΆβˆ’π‘†οŠ¨οŠ¨?

It is known that every function that satisfies 𝑦′′=𝑦 is a combination π‘Žπ‘’+π‘π‘’ο—οŠ±ο— for some constants π‘Ž and 𝑏. Find all functions 𝐢 satisfying 𝐢(0)=5 and πΆβˆ’π‘†=24, where 𝑆=𝐢′.

Sketching graphs of the functions above suggest that 𝐢(π‘₯)=𝐴(π‘₯βˆ’π‘˜)cosh for some constants 𝐴 and π‘˜. Find these constants, where π‘˜>0.

Q2:

The functions 𝑦=π‘’οŠ§ο—, 𝑦=π‘’οŠ¨οŠ±ο—, and 𝑦=π‘’οŠ©οŠ±οŠ¨ο— are three linearly independent solutions of the differential equation 𝑦+2π‘¦β€²β€²βˆ’π‘¦β€²βˆ’2𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=1, 𝑦′(0)=2, and 𝑦′′(0)=0.

Q3:

The functions 𝑦=π‘’οŠ§ο—, 𝑦=π‘’οŠ¨οŠ¨ο—, and 𝑦=π‘’οŠ©οŠ©ο— are three linearly independent solutions of the differential equation π‘¦βˆ’6𝑦′′+11π‘¦β€²βˆ’6𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=0, 𝑦′(0)=0, and 𝑦′′(0)=3.

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