Chapter 8: Problem 18
Identify each of the differential equations as to type (for example, separable, linear first order, linear second order, etc.), and then solve it. \(\left(x \cos y-e^{-\sin y}\right) d y+d x=0\)
Short Answer
Expert verified
The differential equation is separable. After transformation and integration, the solution is \[ \frac {\sin^2 y}{2} + e^{-\sin y} = c \].
Step by step solution
01
Identify the Type of Differential Equation
First, rewrite the given differential equation in the form of \[ \frac{dy}{dx} + A(x,y) = 0 \] The given equation is \[ (x \cos y - e^{-\sin y}) dy + dx = 0 \] Divide the whole equation by \[ (x \cos y - e^{-\sin y}) \] to get \[ dy + \frac{1}{(x \cos y - e^{-\sin y})} dx = 0 \], which is a separable form.
02
Separate the Variables
Rewrite the equation to separate the variables \[ dy = -\frac{1}{(x \cos y - e^{-\sin y})} dx \] Since it's challenging to integrate directly, we should find another approach, realizing it can be transformed into a simpler form.
03
Further Simplify
Rewrite the given equation, emphasizing the separable nature: \[ (x \cos y - e^{-\sin y}) dy = -dx \] This can be written as \[ x \cos y dy - e^{-\sin y} dy = -dx \], which separates variables efficiently but suggests a substitution might simplify integration.
04
Perform Substitution
Let \( u = \sin y \). Then \( du = \cos y \, dy \). The original equation transforms as follows: \[ x du - e^{-u} dy = -dx \], which simplifies integration. The equation becomes: \[ x du - e^{-u} dx = 0 \] or \[ x du = e^{-u} dx \].
05
Integrate Both Sides
Separate the variables: \[ u du = e^{-u} \]. Integrating both sides: \[ \int u \, du = \int e^{-u} \, du \]. Use integration rules: \[ \frac {u^2}{2} = -e^{-u} + C \].
06
Substitute Back
Substitute \(u = \sin y\) back into the equation: \[ \frac {\sin^2 y}{2} + e^{-\sin y} = c \].
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 Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Separable Equations
Separable equations are a type of differential equation where the variables can be separated on opposite sides of the equation. That means, for a differential equation of the form \[ h(y) \, dy = g(x) \, dx. \], we can integrate both sides independently.After separation, we can solve the differential equation by integrating both sides with respect to their respective variables. In our exercise, after dividing by \[ (x \, cos \, y - e^{ -sin \, y}), \], we noticed this separation is possible which makes it a separable differential equation.
It’s a powerful method because it reduces the differential equation to two simpler integrals. This makes it easier to solve compared to directly integrating the mixed terms.
It’s a powerful method because it reduces the differential equation to two simpler integrals. This makes it easier to solve compared to directly integrating the mixed terms.
Substitution Method
The substitution method is a valuable technique to simplify complex differential equations. It allows us to transform the given equation into a more manageable form by introducing a new variable. For example, in our exercise, we used the substitution \[ u = \sin y \] which transformed the original equation.
This approach was necessary because the direct integration was difficult. The substitution method helps in cases where recognizing a pattern or structure in the equation hints that a simpler substitution can simplify the integration process.
After substitution, the resulting differential equation usually becomes easier to handle and integrate.
This approach was necessary because the direct integration was difficult. The substitution method helps in cases where recognizing a pattern or structure in the equation hints that a simpler substitution can simplify the integration process.
After substitution, the resulting differential equation usually becomes easier to handle and integrate.
Integration Techniques
Integration is the process of finding the antiderivative of a function. In solving differential equations, integration techniques are crucial. Basic techniques include:
\[ \int u \, du = \int e^{-u} \ du \]
\[ \frac{u^2}{2} + C = -e^{-u} + C \]
It’s important to correctly apply these rules to solve differential equations as efficiently as possible. Always remember, after integrating, apply the constant of integration (C) to complete the antiderivative.
- Power Rule: \[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
- Exponential Rule: \[ \int e^x dx = e^x + C \]
\[ \int u \, du = \int e^{-u} \ du \]
\[ \frac{u^2}{2} + C = -e^{-u} + C \]
It’s important to correctly apply these rules to solve differential equations as efficiently as possible. Always remember, after integrating, apply the constant of integration (C) to complete the antiderivative.
Linear First Order Equations
Linear first-order differential equations have the general form \[ \frac{dy}{dx} + P(x) y = Q(x). \] They are characterized by the differentials of both x and y being of the first order. These equations always form a straight line in their solution graph. In our exercise, even though the equation \[ \left( x \, \cos y - e^{-\sin y} \right) dy + dx = 0 \] didn’t initially appear linear,
we transformed it into an equivalent separable form \[ dy = - \frac{1}{( x \, \cos y - e^{- \sin y} )} dx, \] followed by substitution to simplify its integration. Linear equations often allow for easier integration contrasts to higher-order or non-linear forms, thanks to their simple y term structure.
we transformed it into an equivalent separable form \[ dy = - \frac{1}{( x \, \cos y - e^{- \sin y} )} dx, \] followed by substitution to simplify its integration. Linear equations often allow for easier integration contrasts to higher-order or non-linear forms, thanks to their simple y term structure.
Transformation of Variables
The transformation of variables method involves changing the variables in a differential equation to simplify and potentially solve it. By introducing a new variable or setting an expression equal to one, we can rearrange and simplify terms that initially appear complex.
In our exercise, by setting \[ u = \sin y \]
We changed the original problem into one involving u and x, which was easier to integrate. This process is also known as change of variables.
Transformation of variables is a technique used in both solving differential equations and integration, making complex equations more manageable and easier to handle.
In our exercise, by setting \[ u = \sin y \]
We changed the original problem into one involving u and x, which was easier to integrate. This process is also known as change of variables.
Transformation of variables is a technique used in both solving differential equations and integration, making complex equations more manageable and easier to handle.