Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 50
Finding a General Solution In Exercises \(43-52\) , use integration to find a general solution of the differential equation. $$\frac{d y}{d x}=2 x \sqrt{4 x^{2}+1}$$
Short Answer
Expert verified
The general solution to the given differential equation is \(y= (1/3) \times (4x^2 + 1)^{3/2} + C\).
Step by step solution
01
Identifying the function to be integrated
The first step is identifying the function that we need to integrate, which is \(f(x) = 2x\sqrt {4x^2 + 1}\) from the original equation.
02
Integration of the function
Next we need to integrate the function \(f(x)\) with respect to \(x\). This would require the use of the substitution method of integration. Let's assume: \(u = 4x^2 + 1\). The derivative of \(u\) with respect to \(x\) would be \(du/dx = 8x\). Also adjusting for the constant, we will create terms such that \(du = 4dx\), implying \(dx = du/(4x)\). Now the integral becomes \(\int 2x \sqrt{u} * du/(4x)\). After simplification of the terms, we get the integral as \(\int 0.5 \sqrt{u} du\).
03
Apply Integration
Apply the integral to the above function. We get \(0.5 * 2/3 * u^{3/2}\). After substitution of \(u = 4x^2 + 1\), \((1/3) \times (4x^2 + 1)^{3/2}\). This is the integral of right-hand side of the function.
04
Solving the differential equation
The general solution of the differential equation is formulated by the integral computed. We get the solution as \(y= (1/3) \times (4x^2 + 1)^{3/2} + C\), where \(C\) represents the constant of integration which can be any constant real number. Therefore, the solution to the differential equation is \(y= (1/3) \times (4x^2 + 1)^{3/2} + C\)
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Methods
Integration is a fundamental concept in calculus, often considered the reverse operation of differentiation. It is used to find areas, volumes, central points, and many useful things. There are several integration methods which are essential for solving calculus problems involving finding the integral of a function. The most common methods include:
Each method has its specific scenarios where it's most effective. For instance, Integration by Substitution is used when a function is the product of a function and its derivative or can be transformed to that form. The purpose of mastering various integration methods is to equip students with the tools to tackle a wide array of functions they may encounter.
- The Power Rule
- Integration by Substitution
- Integration by Parts
- Integration by Partial Fractions
- Trigonometric Integration
- Numerical Integration
Each method has its specific scenarios where it's most effective. For instance, Integration by Substitution is used when a function is the product of a function and its derivative or can be transformed to that form. The purpose of mastering various integration methods is to equip students with the tools to tackle a wide array of functions they may encounter.
Substitution Method of Integration
The substitution method of integration, sometimes called 'u-substitution', is particularly useful when dealing with integrands that are products of a function and its derivative. This method transforms the integral into a simpler form that can be more easily evaluated.
To apply this technique, one identifies a part of the integrand to substitute with a new variable, typically 'u', leading to a new integral in terms of 'u'. The steps include:
Applying this method requires skill in recognizing the function to substitute and the ability to manipulate algebraic expressions efficiently.
To apply this technique, one identifies a part of the integrand to substitute with a new variable, typically 'u', leading to a new integral in terms of 'u'. The steps include:
- Choose a substitution to simplify the integral.
- Express the differential 'dx' in terms of 'du'.
- Perform the integral with respect to 'u'.
- Substitute back in terms of the original variable.
Applying this method requires skill in recognizing the function to substitute and the ability to manipulate algebraic expressions efficiently.
Solving Differential Equations
A differential equation is an equation that relates a function with its derivatives. In comparison to regular equations, differential equations describe how a particular quantity changes over time, rather than providing direct values.
To solve a differential equation, one must find the function that satisfies the equation. This process often involves integration, as in our original exercise where the rate of change of a function was given. The general solution includes an arbitrary constant if it鈥檚 a first-order differential equation or multiple constants for higher-order equations. In the context of our problem, after integrating the given function, we obtained a general solution with a constant 'C', which represents all possible solutions to the differential equation. Solving such equations is crucial in fields ranging from physics and engineering to economics and biology because they model real-world phenomena.
To solve a differential equation, one must find the function that satisfies the equation. This process often involves integration, as in our original exercise where the rate of change of a function was given. The general solution includes an arbitrary constant if it鈥檚 a first-order differential equation or multiple constants for higher-order equations. In the context of our problem, after integrating the given function, we obtained a general solution with a constant 'C', which represents all possible solutions to the differential equation. Solving such equations is crucial in fields ranging from physics and engineering to economics and biology because they model real-world phenomena.
Calculus
Calculus is a branch of mathematics that studies continuous change, and it's divided into two main parts: differential calculus and integral calculus. Differential calculus concerns itself with the concept of a derivative, which represents the rate of change of a quantity.
On the other hand, integral calculus focuses on the concept of the integral, which can be interpreted as the accumulation of quantities and the area under a curve. These two concepts are intertwined by the Fundamental Theorem of Calculus, which connects differentiation with integration. In practice, calculus helps us solve problems in science and engineering by modeling and analyzing dynamic systems, and its applications are vast and critical to technological advancement.
On the other hand, integral calculus focuses on the concept of the integral, which can be interpreted as the accumulation of quantities and the area under a curve. These two concepts are intertwined by the Fundamental Theorem of Calculus, which connects differentiation with integration. In practice, calculus helps us solve problems in science and engineering by modeling and analyzing dynamic systems, and its applications are vast and critical to technological advancement.
