Systems of ODEs: First-Order Linear Equations with Constant Coefficients

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UTT MATH1002 Weeks 3&4 Notes

Systems of ODEs
First-order linear equations with constant coefficients
[pic]
[pic]

Let [pic]

[pic]

Taking Laplace transforms of (1) and (2)

[pic]

[pic]

From (3) and (4)

[pic]

[pic]

We solve this system algebraically for [pic]and [pic] and obtain [pic] by taking inverse transforms.

Example [pic]
[pic]

[pic]

We have

[pic]

[pic]
[pic]

From (5) and (6)

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Since[pic]

[pic]

[pic]

From (II)

[pic]

[pic]

[pic]

[pic]

[pic]

Since[pic]

[pic]

Solution:
[pic]
[pic]

Example

[pic]
Advanced Engineering Mathematics (Kreyszig)

Tank T1 initially contains 100 gal of pure water.
Tank T2 initially contains 100 gal of water in which 150lb of salt are dissolved

The inflow into T1 from outside is 6 gal/min with 6 lb of salt. Obtain the salt content [pic] and [pic] in T1 and T2 respectively. Assume that the mixtures are kept uniform.

[pic]
[pic]

[pic]

Taking Laplace transforms

[pic]
[pic]

[pic]
[pic]

[pic]

By Cramer’s Rule

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Similarly,

[pic]

[pic]

Taking inverse transforms

[pic]
[pic]

Example

[pic][pic]
Advanced Engineering Mathematics (Kreyszig)

The above mechanical system consists of two bodies of mass 1 on 3 springs of the same spring constant k. The masses of the springs are negligible. When the system is in static equilibrium, the masses m1 and m2 are at positions A and B respectively, and the springs all have extension zero as the masses of m1 and m2 are small compared with the spring constant. Each spring is neither extended nor compressed.

With the masses in the positions C and D as shown (above right),

Extension of top spring = [pic]

Extension of middle spring = DC – AB
= (PD – PC) – AB
= [(l1 + l2 + y2) – (l1 + y1)] – l2
= y2 – y1

Consider mass x acc. = Force(mass = 1)

For a stretched spring F = - k x (where k is the spring constant and x is the extension)

[pic]
[pic]

Suppose that the initial conditions are

[pic]

Let [pic]

Taking Laplace transforms

[pic]
[pic]

[pic]
[pic]

These can be solved to give

[pic]

[pic]

[pic]

[pic]

[pic]
[pic]
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