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Chapter 10: Problem 54

The operators \(\mathrm{L}\) and \(\mathrm{M}\) are defined by $$ \mathrm{L}=f_{1}(t) \frac{\mathrm{d}}{\mathrm{d} t}+g_{1}(t) $$ and $$ \mathrm{M}=f_{2}(t) \frac{\mathrm{d}}{\mathrm{d} t}+g_{2}(t) $$ Find expressions for the operators LM and ML. Under what conditions on \(f_{1}, g_{1}, f_{2}\) and \(g_{2}\) is \(\mathrm{LM}=\mathrm{ML}\) ? What conditions do you think linear differential operators must satisfy in order to be commutative?

Short Answer

Expert verified
LM equals ML if \(\frac{\mathrm{d}f_{1}}{\mathrm{d}t} = \frac{f_{1}}{f_{2}} \frac{\mathrm{d}f_{2}}{\mathrm{d}t}\) and \(f_{1}g_{2} = f_{2}g_{1}\).

Step by step solution

01

Calculate LM

To find the operator \(LM\), we need to apply the operator \(M\) followed by \(L\). Start with \(L = f_{1}(t) \frac{\mathrm{d}}{\mathrm{d} t} + g_{1}(t)\). First, apply \(M\):\[ M = \left(f_{2}(t) \frac{\mathrm{d}}{\mathrm{d} t} + g_{2}(t)\right) \]. Now calculate \(LM\):\[(LM)(u) = L(M(u)) = L\left(f_{2}(t) \frac{\mathrm{d}u}{\mathrm{d}t} + g_{2}(t)u \right)\]Apply \(L\):\[= f_{1}(t) \frac{\mathrm{d}}{\mathrm{d}t} \left( f_{2}(t) \frac{\mathrm{d}u}{\mathrm{d}t} + g_{2}(t)u \right) + g_{1}(t)\left( f_{2}(t) \frac{\mathrm{d}u}{\mathrm{d}t} + g_{2}(t)u \right)\]Use the product rule on \( f_{1}(t) \frac{\mathrm{d}}{\mathrm{d}t} \):\[= f_{1}(t)\left(\frac{\mathrm{d}f_{2}}{\mathrm{d}t}\frac{\mathrm{d}u}{\mathrm{d}t} + f_{2}(t)\frac{\mathrm{d}^2 u}{\mathrm{d}t^2} + \frac{\mathrm{d}g_{2}}{\mathrm{d}t}u + g_{2}(t)\frac{\mathrm{d}u}{\mathrm{d}t}\right) + g_{1}(t)\left(f_{2}(t)\frac{\mathrm{d}u}{\mathrm{d}t} + g_{2}(t)u\right)\].
02

Calculate ML

Now calculate the operator \(ML\), which is applying \(L\) followed by \(M\):\[ (ML)(u) = M(L(u)) = M\left(f_{1}(t) \frac{\mathrm{d}u}{\mathrm{d}t} + g_{1}(t)u \right)\]Apply \(M\):\[= f_{2}(t) \frac{\mathrm{d}}{\mathrm{d}t} \left( f_{1}(t) \frac{\mathrm{d}u}{\mathrm{d}t} + g_{1}(t)u \right) + g_{2}(t)\left( f_{1}(t) \frac{\mathrm{d}u}{\mathrm{d}t} + g_{1}(t)u \right)\]Use the product rule on \( f_{2}(t)\frac{\mathrm{d}}{\mathrm{d}t} \):\[= f_{2}(t)\left(\frac{\mathrm{d}f_{1}}{\mathrm{d}t}\frac{\mathrm{d}u}{\mathrm{d}t} + f_{1}(t)\frac{\mathrm{d}^2 u}{\mathrm{d}t^2} + \frac{\mathrm{d}g_{1}}{\mathrm{d}t}u + g_{1}(t)\frac{\mathrm{d}u}{\mathrm{d}t}\right) + g_{2}(t)\left(f_{1}(t)\frac{\mathrm{d}u}{\mathrm{d}t} + g_{1}(t)u\right)\].
03

Set Conditions for LM = ML

To determine when \(LM = ML\), compare the expressions from Steps 1 and 2. Simplify both expressions to compare terms.Both operators are equal if the cross terms and constant terms match. Hence, set the coefficients of all derivative terms equal:\[ f_{1}(t) \frac{\mathrm{d}f_{2}}{\mathrm{d}t} = f_{2}(t) \frac{\mathrm{d}f_{1}}{\mathrm{d}t} \]\[ f_{1}(t)g_{2}(t) = f_{2}(t)g_{1}(t) \].The commutative condition is satisfied if these two conditions are true.
04

Conclusion on Commutativity

Linear differential operators are commutative if they satisfy the computed conditions: the derivatives of the functions \(f_{1}(t), f_{2}(t)\) must be proportional, and the cross-multiplied \(g\) terms must agree. Therefore, if \(f_{1}, g_{1}, f_{2}, \) and \(g_{2}\) satisfy these conditions, \(LM = ML\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Commutativity in Differential Operators
Commutativity is all about the order in which operations are executed. For two operators to be commutative, applying them in different sequences should yield the same outcome. In simpler terms, for operators \(L\) and \(M\), they are commutative if \(LM = ML\).

Considering the given operators:
  • \(L=f_{1}(t) \frac{\mathrm{d}}{\mathrm{d} t} + g_{1}(t)\)
  • \(M=f_{2}(t) \frac{\mathrm{d}}{\mathrm{d} t} + g_{2}(t)\)
The goal is to find conditions under which these operators satisfy \(LM = ML\). Through calculation, it's found that:
  • The derivatives for \(f_{1}(t)\) and \(f_{2}(t)\) must be proportional.
  • It must hold that \(f_{1}(t)g_{2}(t) = f_{2}(t)g_{1}(t)\).
When these conditions are met, the order in which you apply \(L\) and \(M\) does not affect the result.
Introduction to Linear Differential Equations
Linear differential equations are fundamental tools in mathematics, modeling a wide range of phenomena. They are equations that involve functions and their derivatives, with coefficients dependent on the independent variable鈥攊n this instance, \(t\).

Linear here means that these equations do not involve products or nonlinear terms of the unknown function or its derivatives. Given our operators
  • \(L = f_{1}(t) \frac{\mathrm{d}}{\mathrm{d} t} + g_{1}(t)\)
  • \(M = f_{2}(t) \frac{\mathrm{d}}{\mathrm{d} t} + g_{2}(t)\)
Each operator acts linearly on a function \(u(t)\), making both \(L\) and \(M\) linear differential operators.

These concepts lead to understanding properties, such as commutativity, under specific conditions of linear operators. The approach to solving these equations often involves the use of methods that respect the linear structure, ensuring easier manipulation and interpretation of solutions.
The Product Rule in Calculus and Differential Operators
The product rule is a crucial rule in calculus. It explains how to differentiate the product of two functions. The rule can be stated simply: \((fg)' = f'g + fg'\).

When dealing with differential operators, especially with expressions like \(LM\) or \(ML\), it's crucial to use the product rule correctly. This is because these operators involve applying derivatives sequentially, which requires careful handling of functions involved.

For instance:
  • When calculating \(LM\), the product rule needs to be applied inside the expression \(L(M(u))\).
  • Similarly, when calculating \(ML\), apply the product rule within \(M(L(u))\).
In our exercises, each step involving product rule shows how to distribute differentiation across terms like \(f_{1}(t) \frac{\mathrm{d}f_{2}(t)}{\mathrm{d}t}\) or \(f_{2}(t) \frac{\mathrm{d}f_{1}(t)}{\mathrm{d}t}\).

This meticulous application of the product rule is what allows us to discover the specific conditions under which differential operators would commute. Mastering the product rule hence not only aids in solving differential equations but is integral in understanding the behavior of such operators.

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Most popular questions from this chapter

Find the damping parameters and natural frequencies of the systems governed by the following second-order linear constant-coefficient differential equations: (a) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+7 \frac{\mathrm{d} x}{\mathrm{~d} t}+2 x=0\) (b) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+p \frac{\mathrm{d} x}{\mathrm{~d} t}+p^{1 / 2} x=0\) (c) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 a q \frac{\mathrm{d} x}{\mathrm{~d} t}+\frac{1}{2} q x=0\) (d) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+14 \frac{\mathrm{d} x}{\mathrm{~d} t}+2 \alpha x=0\)

Sketch the direction field of the differential equation $$ \frac{\mathrm{d} x}{\mathrm{~d} t}=-2 t $$ Find the solution of the equation. Sketch the particular solutions for which \(x(0)=2\), and for which \(x(2)=-3\), and check that these are consistent with your direction field.

Classify the following differential equations as linear homogeneous, linear nonhomogeneous or nonlinear differential equations, state their order and name the dependent and independent variables. (a) \(\frac{\mathrm{d}^{2} p}{\mathrm{~d} z^{2}} \frac{\mathrm{d} p}{\mathrm{~d} z}+(\sin z) p=\ln z\) (b) \(\frac{\mathrm{d}^{2} s}{\mathrm{~d} t^{2}}+(\sin t) \frac{\mathrm{d} s}{\mathrm{~d} t}+(t+\cos t) s=\mathrm{e}^{t}\) (c) \(\left(\frac{\mathrm{d}^{3} p}{\mathrm{~d} y^{3}}\right)^{1 / 2}+4 \frac{\mathrm{d}^{2} p}{\mathrm{~d} y^{2}}-6 \frac{\mathrm{d} p}{\mathrm{~d} y}+8 p=0\) (d) \(\frac{\mathrm{d} r}{\mathrm{~d} z}+z^{2}=0\) (e) \(\frac{\mathrm{d} x}{\mathrm{~d} t}=f(t) x\) (f) \(\frac{\mathrm{d} x}{\mathrm{~d} t}=f(t) x+g(t)\) (g) \(\frac{\mathrm{d}^{3} p}{\mathrm{~d} q^{3}}+\frac{\mathrm{d}^{2} p}{\mathrm{~d} q^{2}} p+4 q^{2}=0\) (h) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} y^{2}}=\frac{y}{x^{2}-1}\) (i) \((\sin z) \frac{\mathrm{d} y}{\mathrm{~d} z}+\frac{\cos z}{z} y=0\)

For each of the following sets of linearly dependent functions find \(k_{1}, k_{2}, \ldots\) such that \(k_{1} f_{1}+k_{2} f_{2}+\ldots=0\) (a) \(\\{t+1, t, 2\\}\) (b) \(\left\\{t^{2}-1, t^{2}+1, t-1, t+1\right\\}\)

Solve the following differential equation problems: (a) \(\frac{\mathrm{d} x}{\mathrm{~d} t}-2 x=t, \quad x(0)=2\) (b) \(\frac{\mathrm{d} x}{\mathrm{~d} t}+2 t x=\left(t-\frac{1}{2}\right) \mathrm{e}^{-t}, \quad x(0)=1\) (c) \(\frac{\mathrm{d} x}{\mathrm{~d} t}+3 x=\mathrm{e}^{2 t}, \quad x(0)=2\) (d) \(\frac{\mathrm{d} x}{\mathrm{~d} t}+x \sin t=\sin t, \quad x(\pi)=\mathrm{e}\)
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