Reduction of order

Key takeaways

Reduction of order belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Reduction of order to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Reduction of order from memory before moving on to harder problems.

Reference excerpt

Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y 1 ( x ) {\displaystyle y_{1}(x)} is known and a second linearly independent solution y 2 ( x ) {\displaystyle y_{2}(x)} is desired. The method also applies to n-th order equations. In this case the ansatz will yield an (n−1)-th order equation for v {\displaystyle v} .

Second-order linear ordinary differential equations

An example Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE)

a y ″ ( x ) + b y ′ ( x ) + c y ( x ) = 0 , {\displaystyle ay''(x)+by'(x)+cy(x)=0,}

where a , b , c {\displaystyle a,b,c} are real non-zero coefficients. Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, b 2 − 4 a c {\displaystyle b^{2}-4ac} , vanishes. In this case,

a y ″ ( x ) + b y ′ ( x ) + b 2 4 a y ( x ) = 0 , {\displaystyle ay''(x)+by'(x)+{\frac {b^{2}}{4a}}y(x)=0,}

from which only one solution,

y 1 ( x ) = e − b 2 a x , {\displaystyle y_{1}(x)=e^{-{\frac {b}{2a}}x},}

can be found using its characteristic equation. The method of reduction of order is used to obtain a second linearly independent solution to this differential equation using our one known solution. To find a second solution we take as a guess

y 2 ( x ) = v ( x ) y 1 ( x ) {\displaystyle y_{2}(x)=v(x)y_{1}(x)}

where v ( x ) {\displaystyle v(x)} is an unknown function to be determined. Since y 2 ( x ) {\displaystyle y_{2}(x)} must satisfy the original ODE, we substitute it back in to get

a ( v ″ y 1 + 2 v ′ y 1 ′ + v y 1 ″ ) + b ( v ′ y 1 + v y 1 ′ ) + b 2 4 a v y 1 = 0. {\displaystyle a\left(v''y_{1}+2v'y_{1}'+vy_{1}''\right)+b\left(v'y_{1}+vy_{1}'\right)+{\frac {b^{2}}{4a}}vy_{1}=0.}

Rearranging this equation in terms of the derivatives of v ( x ) {\displaystyle v(x)} we get

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Reduction of order

Start with the simplest possible case. Write down what Reduction of order claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Reduction of order before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Reduction of order ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Reduction of order

In research: Reduction of order appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Reduction of order in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Reduction of order is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ordinary differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Reduction of order outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Reduction of order in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Reduction of order means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Reduction of order out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Reduction of order in simple terms?

Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y 1 ( x ) {\displaystyle y_{1}(x)} is known and a second linearly independent solution y 2 ( x ) {\displaystyle y_{2}(x)} is d…

Why does Reduction of order matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Reduction of order?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Reduction of order.