Chapter 4: Problem 78
Solve the differential equation. \(\frac{d y}{d x}=\frac{1-2 x}{4 x-x^{2}}\)
Short Answer
Expert verified
The solution to the differential equation is \(y = A\frac{1-2x}{4 x-x^{2}}\), where \(A\) is any arbitrary constant.
Step by step solution
01
Rewrite the differential equation
The first step is to isolate \(dy\) and \(dx\). To do this, you must acknowledge the differential equation as a product of two functions: \(dy = \frac{1-2x}{4x-x^2}dx\).
02
Separate the variables
The next step is to group all \(y\)-terms and \(dy\) on one side and all \(x\)-terms and \(dx\) on the other. Since there is only one \(y\)-term on the left side of the differential equation, it can be rewritten as: \(dy = \frac{1}{4 x-x^{2}}dx(-2x + 1)\)
03
Integrate both sides of the equation
Perform a definite integral on both sides of the equation and solve: \(\int dy = \int \frac{1-2x}{4 x-x^{2}}dx\). The right side can be seen as a simple substitution \(u = 4x - x^2\), \(du = -2x + 1\). Therefore, the integration becomes \(\int du/u\) which solves to \(\ln |u| + C\), where C is the integration constant.
04
Substitute u back in for x
Substitute \(u\) back in for \(x\), to get the solution to the differential equation. This will give: \(y = \ln |\frac{1-2x}{4x-x^2}| + C\)
05
Exponentiate both sides to solve for y
Both sides are exponentiated to make it easier to solve for \(y\). This gives the solution as: \(y = e^{\ln |\frac{1-2x}{4 x-x^{2}}|} e^C\), or equivalently, \(y = A\frac{1-2x}{4 x-x^{2}}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Separation of Variables
Separation of Variables is a common method for solving ordinary differential equations (ODEs), particularly when dealing with variables that can be separated onto different sides of the equation. The principle is guided by rearranging the equation so that each variable and its derivative are isolated on opposite sides.
For example, in the provided exercise, the differential equation is given as \(\frac{d y}{d x}=\frac{1-2 x}{4 x-x^{2}}\). The goal is to rewrite it so you have all instances of \(y\) and \(dy\) on one side, and all the \(x\) terms and \(dx\) on the other. The equation is already in an almost separated form, making it simpler to apply this technique:
\(dy = \frac{1-2x}{4x-x^2}dx\).
Successful use of this method reduces the complex problem into a more manageable form, making it possible to use integration to find the function \(y\). This is particularly valuable for students as it provides a procedural approach to solving differential equations.
For example, in the provided exercise, the differential equation is given as \(\frac{d y}{d x}=\frac{1-2 x}{4 x-x^{2}}\). The goal is to rewrite it so you have all instances of \(y\) and \(dy\) on one side, and all the \(x\) terms and \(dx\) on the other. The equation is already in an almost separated form, making it simpler to apply this technique:
\(dy = \frac{1-2x}{4x-x^2}dx\).
Successful use of this method reduces the complex problem into a more manageable form, making it possible to use integration to find the function \(y\). This is particularly valuable for students as it provides a procedural approach to solving differential equations.
Integration of Differential Equations
Integration is the inverse operation to differentiation and plays a pivotal role in solving differential equations. In the context of solving such equations, integration acts as a tool to reconstruct the original function from its derivative.
During the step-by-step solution, after the variables are separated, integration is applied to both sides of the equation independently. For instance, in the given equation, after the separation of variables, you have an integrable equation:
\(\int dy = \int \frac{1-2x}{4 x-x^{2}}dx\).
The integration of the right side requires a technique called substitution, which simplifies the integrand to a recognizable form that can be easily integrated. In this case, setting \(u = 4x - x^2\) allows us to rewrite and integrate the equation with respect to \(u\).
Understanding this integral conversion and the execution of integration is critical, as it aids in constructing a solution that describes the behavior of the dependent variable in terms of the independent variable.
During the step-by-step solution, after the variables are separated, integration is applied to both sides of the equation independently. For instance, in the given equation, after the separation of variables, you have an integrable equation:
\(\int dy = \int \frac{1-2x}{4 x-x^{2}}dx\).
The integration of the right side requires a technique called substitution, which simplifies the integrand to a recognizable form that can be easily integrated. In this case, setting \(u = 4x - x^2\) allows us to rewrite and integrate the equation with respect to \(u\).
Understanding this integral conversion and the execution of integration is critical, as it aids in constructing a solution that describes the behavior of the dependent variable in terms of the independent variable.
Definite Integral Calculus
Definite integral calculus involves calculating the area under a curve within a certain interval and is a fundamental concept in various fields involving mathematics. In our context, 'definite integral' may be somewhat of a misnomer as we often deal with the indefinite integration when solving differential equations without initial conditions.
However, the process shares some commonalities with definite integration, such as finding the antiderivative. In the problem provided, the indefinite integral \(\int dy = \int \frac{1-2x}{4 x-x^{2}}dx\) results in a natural logarithm function when integrated, represented as \(\ln |u|\). This process mirrors finding the area under the curve of the function \(\frac{1}{u}\) from some point to another. Typically, a definite integral would have limits of integration, but here the 'constant of integration' \(C\) represents the family of solutions that describe the original differential equation.
Toric words, the concepts behind definite and indefinite integrals are essential as they both represent the accumulation of quantities, which is the essence of integration. The understanding of these principles allows students to solve a wide array of problems within calculus and beyond.
However, the process shares some commonalities with definite integration, such as finding the antiderivative. In the problem provided, the indefinite integral \(\int dy = \int \frac{1-2x}{4 x-x^{2}}dx\) results in a natural logarithm function when integrated, represented as \(\ln |u|\). This process mirrors finding the area under the curve of the function \(\frac{1}{u}\) from some point to another. Typically, a definite integral would have limits of integration, but here the 'constant of integration' \(C\) represents the family of solutions that describe the original differential equation.
Toric words, the concepts behind definite and indefinite integrals are essential as they both represent the accumulation of quantities, which is the essence of integration. The understanding of these principles allows students to solve a wide array of problems within calculus and beyond.
