Name: modeling motion with second-order equations
Uploaded: 2026-01-12T12:25:00.000Z
Duration: 1 min 20 s
Channel: JoVE

Diferansiyel denklem, bilinmeyen bir fonksiyon ve bir veya daha fazla türevini içeren bir denklemdir. Bir diferansiyel denklemin mertebesi, denklemde bulunan en yüksek dereceden türevle verilir.Denklem sadece birinci türevi içerdiğinde, birinci mertebe denir. İkinci türevi içeriyorsa, ikinci dereceden denir ve böyle devam eder.Örneğin, bir yay-kütle sistemi, kütle üzerindeki net kuvvet ile yaydaki geri kazandırıcı kuvveti eşitleyen basit bir harmonik osilatör olarak modellenebilir; bu, doğrusal yaylar için Hooke yasasıyla tanımlanır.Yay kuvveti, yer değiştirmenin ters yönünde etki eder ve kütlenin denge konumundan ne kadar uzaklaştığıyla orantılıdır.Newton'un ikinci yasası, kütle üzerindeki net kuvveti onun ivmelenmesine bağlar. Bu iki ilişkiyi birleştirdiğimizde, kütle ile ivmelenme sayısının negatif sabit çarpan yer değiştirmeye eşit olduğu bir denklem elde edilir.Hızlanma, zaman içindeki yer değiştirmenin ikinci türevi olarak yeniden yazılıp, ikinci dereceden diferansiyel denklem elde edilir.Bu ikinci türev, sistemin hareketini diferansiyel denkleme bağlar.

diferansiyel denklemlere giriş

Bir diferansiyel denklem, bir fonksiyon ile onun t&#xFC;revleri aras&#x131;nda ili&#x15F;ki kuran matematiksel bir ifadedir. Bu denklemler, bilimin ve m&#xFC;hendisli&#x11F;in &#xE7;e&#x15F;itli alanlar&#x131;nda dinamik sistemlerin modellenmesinde temel &#xF6;neme sahiptir. Bir diferansiyel denklemin mertebesi, denklemde yer alan en y&#xFC;ksek mertebeden t&#xFC;rev taraf&#x131;ndan belirlenir. Birinci mertebeden bir diferansiyel denklem yaln&#x131;zca birinci t&#xFC;revi i&#xE7;erirken, ikinci mertebeden bir diferansiyel denklem bilinmeyen fonksiyonun ikinci t&#xFC;revine kadar olan t&#xFC;revlerini i&#xE7;erir.Fiziksel Sistemlerde &#x130;kinci Mertebeden Diferansiyel Denklemler&#x130;kinci mertebeden bir diferansiyel denklemin klasik bir &#xF6;rne&#x11F;i, yay&#x2013;k&#xFC;tle sisteminin hareketinin modellenmesinde ortaya &#xE7;&#x131;kar. Bu idealize edilmi&#x15F; sistemde, Hooke Yasas&#x131;&#x2019;na uyan bir yaya bir k&#xFC;tle ba&#x11F;lanm&#x131;&#x15F;t&#x131;r ve sistem herhangi bir s&#xF6;n&#xFC;mleme ya da d&#x131;&#x15F; kuvvet etkisi olmaks&#x131;z&#x131;n kabul edilir. K&#xFC;tle denge konumundan uzakla&#x15F;t&#x131;r&#x131;ld&#x131;&#x11F;&#x131;nda yay, yer de&#x11F;i&#x15F;tirmeyle do&#x11F;rusal olarak orant&#x131;l&#x131; olan ve yer de&#x11F;i&#x15F;tirmeye z&#x131;t y&#xF6;nde etki eden bir geri &#xE7;a&#x11F;&#x131;r&#x131;c&#x131; kuvvet uygular. Matematiksel olarak, bu geri &#xE7;a&#x11F;&#x131;r&#x131;c&#x131; kuvvet a&#x15F;a&#x11F;&#x131;daki gibi verilir:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:11.24%;"><img style="aspect-ratio:78/32;" src="https://cloudfront.jove.com/files/ftp_upload/40597/40597_001_editor_40597.svg" alt="40597_001.svg" /></figure>Burada k yay sabitini, x ise denge konumuna g&#xF6;re yer de&#x11F;i&#x15F;tirmeyi ifade eder.Newton&#x2019;un ikinci yasas&#x131;na g&#xF6;re, bir cisme etki eden net kuvvet, cismin k&#xFC;tlesi ile ivmesinin &#xE7;arp&#x131;m&#x131;na e&#x15F;ittir. Yay kuvveti Newton&#x2019;un yasas&#x131;na yerine yaz&#x131;ld&#x131;&#x11F;&#x131;nda a&#x15F;a&#x11F;&#x131;daki ifade elde edilir:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:12.27%;"><img style="aspect-ratio:81/44;" src="https://cloudfront.jove.com/files/ftp_upload/40597/40597_002_editor_40597.svg" alt="40597_002.svg" /></figure>&#x130;vme, yer de&#x11F;i&#x15F;tirmenin zamana g&#xF6;re ikinci t&#xFC;revi olarak ifade edildi&#x11F;inde.Terimlerin yeniden d&#xFC;zenlenmesi, ikinci mertebeden do&#x11F;rusal diferansiyel denklemin standart formunu verir:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:15.57%;"><img style="aspect-ratio:148/47;" src="https://cloudfront.jove.com/files/ftp_upload/40597/40597_003_editor_40597.svg" alt="40597_003.svg" /></figure>Bu denklem basit harmonik hareketi a&#xE7;&#x131;klar ve mekanik titre&#x15F;imler ile sal&#x131;n&#x131;ml&#x131; sistemlerin incelenmesinde temel bir model niteli&#x11F;i ta&#x15F;&#x131;r.

Diferansiyel Denklemlere Giriş

Bir diferansiyel denklem, bir fonksiyon ile onun t&#xFC;revleri aras&#x131;nda ili&#x15F;ki kuran matematiksel bir ifadedir. Bu denklemler, bilimin ve ...

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Calculus

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Unit

Birim

Differential Equations

Diferansiyel Denklemler

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

Introduction to Differential Equations

modeling motion with second-order equations

Differential equations connect a function with its derivatives. They are used to describe changing systems in science and engineering. A differential equation is classified by order, which is the highest derivative it contains. A first-order equation includes only the first derivative, while a second-order equation includes the second derivative of the unknown function.

A common second-order differential equation comes from a spring-mass system. In this idealized model, a mass is attached to a spring that follows Hooke’s Law and has no damping or external forces. When the mass moves away from equilibrium, the spring pulls it back. The restoring force is proportional to the displacement and points in the opposite direction.

Newton’s second law links that force to motion. It states that net force equals mass times acceleration. When the spring force is substituted into Newton’s law, and acceleration is written as the second derivative of displacement with respect to time, the result is the standard second-order linear differential equation for the system. This equation describes simple harmonic motion and is a key model for mechanical vibrations and oscillating systems.

Modeling Motion with Second-Order Equations

modeling-motion-with-second-order-equations

introduction to differential equations

Differential equations connect a function with its derivatives. They are used to describe changing systems in science and engineering. A differential equation is classified by order, which is the highest derivative it contains. A first-order equation includes only the first derivative, while a second-order equation includes the second derivative of the unknown function. A common second-order differential equation comes from a spring-mass system. In this idealized model, a mass is attached to a...

modeling with differential equations

Population Growth: Exponential vs. Logistic

Population growth can be modeled with differential equations by treating population size, P(t), as a function of time. The derivative of P(t) shows how fast the population changes. A simple starting idea is that the growth rate is proportional to the population size. That assumption gives an exponential growth model. In this model, the population rises continuously and faster over time. It can be a useful first approximation, but it does not match long-term behavior in nature. Real...

separable differential equations

Solving Separable Differential Equations

A separable differential equation is a first-order differential equation that can be written with dy/dx as a product of one function of x and one function of y. This form makes it possible to separate the variables so that all y terms stay on one side and all x terms stay on the other. After the variables are separated, each side can be integrated with respect to its own variable. The result is a relationship between x and y. Depending on the functions involved, that relationship may be...

orthogonal trajectories

Perpendicular Curves in Calculus

Orthogonal trajectories are a pair of curve families that cross at right angles. In this example, the starting family is made of parabolas that open sideways along the x-axis. They share the same shape, but a scaling parameter changes how wide each curve is. Every curve in the family passes through the origin and spreads out at a different rate. To find the orthogonal trajectories, the slope of each parabola must be calculated first. The slope of a tangent shows how steep the curve is at a...

linear differential equations

Integrating Factors for Motion with Drag

The integrating factor method solves first-order linear differential equations that are not separable. It is especially useful when a system has a constant driving force and a resistive force that depends on the changing variable. In these problems, the equation can be rearranged so the left-hand side becomes a product rule after multiplication by an integrating factor. A clear example is a car moving under a constant engine force while air resistance acts against its motion. Newton’s second...

differential equations: problem solving

Differential Equations and Falling Motion

Differential equations can model the motion of a falling object when gravity and air resistance act at the same time. In this example, a heavy test weight is released from rest during a ship safety check. Gravity pulls the weight downward, and air resistance pushes upward as the speed increases. Newton’s Second Law links the net force on the weight to its acceleration. Gravity provides a constant downward force equal to mass times gravitational acceleration. Air resistance is modeled as a...

key terms and concepts

Learn how differential equations describe changing systems and how a spring-mass model leads to a second-order equation for simple harmonic motion.

Modeling Motion with Second-Order Equations | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Modeling Motion with Second-Order Equations

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