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Chapter 4: Problem 7

Find a first order differential equation for the given family of curves. $$ y=\sin x+c e^{x} $$

Short Answer

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#Question: Find a first-order differential equation representing the given family of curves: $$y = \sin x + ce^x$$ #Answer: The first-order differential equation representing the given family of curves is: $$\frac{dy}{dx} = y + \cos x - \sin x$$

Step by step solution

01

Taking the derivative of the curve function with respect to x

First, let's find the derivative of the given function with respect to x. Recall that the derivatives of sin(x) and e^x are cos(x) and e^x, respectively. Given function is: $$ y = \sin x + ce^x $$ Now, taking the first derivative of y with respect to x: $$ \frac{dy}{dx} = \cos x + ce^x $$
02

Rearranging the equation to get a relationship between y, x, and their derivatives

We are now going to create a relationship between y, x, and their derivatives. We already found out that: $$ \frac{dy}{dx} = \cos x + ce^x $$ However, we can see that the second term contains 'c' multiplied by e^x, which is already present in the given function y. So, let's write y in terms of this term: $$ y = \sin x + ce^x \implies ce^x = y - \sin x $$ Now, substitute the value of ce^x from the equation above into the derivative expression: $$ \frac{dy}{dx} = \cos x + (y - \sin x) $$ And finally, the first-order differential equation representing the given family of curves is: $$ \frac{dy}{dx} = y + \cos x - \sin x $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Order Differential Equation
A first order differential equation involves derivatives of a function along with the function itself. Specifically, it includes only the first derivative (rate of change) of a function, and is helpful for modeling various physical phenomena.
  • These types of equations can often describe how a system evolves over time or under specific conditions.
  • The general form of a first order differential equation can be expressed as \( \frac{dy}{dx} = f(x, y) \), where \( f(x, y) \) is a function of both \( x \) and \( y \).
In the context of the problem, we derived such an equation that captures the behavior of a family of curves defined by the original problem function.
Family of Curves
The family of curves is essentially a set of curves defined by a common functional form, but with different specific parameters. In our case, the family is represented by the expression \( y = \sin x + c e^x \), where "\( c \)" is a constant.
  • The parameter \( c \) changes the shape or position of each curve in the family.
  • This commonality, such as the functional form, allows for analysis that applies to any curve described by the family.
Finding a differential equation that applies to this family can ultimately describe the dynamic relationship between the different parameters involved (like \( c \), \( x \), and \( y \)).
Derivatives
Derivatives represent the rate of change of a function with respect to a variable, often denoted by \( \frac{dy}{dx} \) for functions of \( y \) with respect to \( x \).
  • In the exercise, we calculated the first derivative of \( y = \sin x + c e^x \) to be \( \cos x + ce^x \).
  • This process follows using derivative rules: the derivative of \( \sin x \) is \( \cos x \) and the derivative of \( e^x \) is \( e^x \).
Calculating derivatives is essential as they inform us about the instantaneous rate of change of one variable relative to another, a fundamental concept in differential equations.
Integration
Integration is the reverse process of differentiation. While derivatives are about the rate of change, integration deals with the accumulation and summing up of quantities over intervals.
  • Though not directly used in deriving the differential equation in this exercise, integration is an integral part of working with differential equations.
  • In solving differential equations, integration helps find the original function from its derivative.
  • It's commonly used to find solutions containing "+ C", the constant of integration, indicating the family of solutions.
Understanding integration is crucial to fully grasp how differential equations work, as it provides the means to revert from a derivative to the complete expression of the original function.

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Most popular questions from this chapter

Consider the mixing problem of Example 4.2.4 in a tank with infinite capacity, but without the assumption that the mixture is stirred instantly so that the salt is always uniformly distributed throughout the mixture. Assume instead that the distribution approaches uniformity as \(t \rightarrow \infty .\) In this case the differential equation for \(Q\) is of the form $$ Q^{\prime}+\frac{a(t)}{t+100} Q=1 $$ where \(\lim _{t \rightarrow \infty} a(t)=1\) (a) Let \(K(t)\) be the concentration of salt at time \(t\). Assuming that \(Q(0)=Q_{0},\) can you guess the value of \(\lim _{t \rightarrow \infty} K(t) ?\) (b) Use numerical methods to confirm your guess in the these cases: (i) \(a(t)=t /(1+t)\) $$ \text { (ii) } a(t)=1-e^{-t^{2}} $$ (iii) \(a(t)=1+\sin \left(e^{-t}\right)\).

Let \(p=p(t)\) be the quantity of a product present at time \(t\). The product is manufactured continuously at a rate proportional to \(p\), with proportionality constant \(1 / 2,\) and it's consumed continuously at a rate proportional to \(p^{2}\), with proportionality constant \(1 / 8\). Find \(p(t)\) if \(p(0)=100\).

A firefighter who weighs \(192 \mathrm{lb}\) slides down an infinitely long fire pole that exerts a frictional resistive force with magnitude proportional to her speed, with constant of proportionality \(k\). Find \(k\), given that her terminal velocity is \(-16 \mathrm{ft} / \mathrm{s}\), and then find her velocity \(v\) as a function of \(t\). Assume that she starts from rest.

A savings account pays \(5 \%\) per annum interest compounded continuously. The initial deposit is \(Q_{0}\) dollars. Assume that there are no subsequent withdrawals or deposits. (a) How long will it take for the value of the account to triple? (b) What is \(Q_{0}\) if the value of the account after 10 years is $$\$ 100,000$$ dollars?

Control mechanisms allow fluid to flow into a tank at a rate proportional to the volume \(V\) of fluid in the tank, and to flow out at a rate proportional to \(V^{2}\). Suppose \(V(0)=V_{0}\) and the constants of proportionality are \(a\) and \(b,\) respectively. Find \(V(t)\) for \(t>0\) and find \(\lim _{t \rightarrow \infty} V(t)\).
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