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Chapter 2: Problem 2

In Problems \(1-6,\) determine whether the given differential equation is separable. $$ \frac{d y}{d x}=4 y^{2}-3 y+1 $$

Short Answer

Expert verified
The given differential equation is not separable.

Step by step solution

01

Check if the equation is separable

The differential equation given is \(\frac{d y}{d x}=4 y^{2}-3 y+1\). A first glance at the problem shows that the right side of the given equation can't be factored into a product of functions of y and x separately, thus it seems that the equation is not separable. However, we need to explore a little more to confirm this.
02

Try expressing the equation in the form \( f(y)dy = g(x)dx \)

An equation can't be called non-separable just by looking, there might be some way to rewrite it such that it becomes separable. However, in this case, there isn't any x term in the given equation, so we cannot express it as a function of x times dx equals a function of y times dy. It confirms our initial judgment that the equation is indeed non-separable.

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Key Concepts

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

Ordinary Differential Equations (ODEs)
Ordinary differential equations (ODEs) are equations that involve functions and their derivatives.
These equations express the relationship between a function and its rate of change. ODEs are crucial in modeling various phenomena in science and engineering, such as motion, heat, and fluid flow.
  • **Order of an ODE:** The order of an ordinary differential equation is determined by the highest derivative present. If the highest derivative is the first derivative, it's a first-order ODE; if the highest is the second derivative, it's a second-order ODE, and so on.
  • **Linear vs. Nonlinear:** ODEs can be linear or nonlinear. Linear ODEs have solutions that can be expressed as linear combinations of the unknown functions and their derivatives, whereas nonlinear ODEs involve products or powers of the unknown functions.
  • **Initial Value Problems (IVP):** An IVP for an ODE includes conditions specified at a single point, which helps determine a unique solution among many potential ones.
Understanding ODEs is essential for interpreting the behavior of dynamic systems and for effectively applying mathematical models to real-world problems.
Factoring Expressions
Factoring is a mathematical process used to break down expressions into simpler components, or 'factors,' that, when multiplied together, give the original expression.
This technique is often used to simplify problems and make them easier to solve. It plays a crucial role in algebra and calculus related tasks.
  • **Common Factoring Techniques:** Factoring can involve methods like pulling out a greatest common factor (GCF), applying the difference of squares, or factoring trinomials into binomial products.
  • **Importance in Differential Equations:** In the context of differential equations, factoring can help to transform equations into a solvable format, particularly when looking for solutions to linear equations or seeking to determine if an equation is separable.
While the process of factoring is straightforward in many cases, some expressions may be complex enough to appear non-factorable at first glance, necessitating deeper analysis or advanced techniques.
Non-separable Equations
Non-separable differential equations are those that cannot be written in the form where each variable can be placed on separate sides of the equation, like \(f(y)dy = g(x)dx\).
In other words, they cannot be expressed as the product of two functions, one depending solely on \(x\) and the other solely on \(y\).
  • **Characteristics:** These equations contain terms that mix derivatives and the function itself in a way that precludes simple separation. Non-separable equations challenge simple integration techniques, often requiring more complex methods for solving, such as numerical approaches or different representation strategies.
  • **Why Non-separable?:** In our example, \(\frac{dy}{dx}=4y^2-3y+1\), there's no straightforward way to split the terms regarding \(x\) and \(y\). It's not possible to arrange the expression into separate multiplicative parts involving \(dx\) and \(dy\).
These equations highlight situations where interrelations between variables are too intertwined for straightforward separation, needing more intricate problem-solving strategies.

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Most popular questions from this chapter

Free Fall. In Section 2.1, we discussed a model for an object falling toward Earth. Assuming that only air resistance and gravity are acting on the object, we found that the velocity \( \boldsymbol{v} \) must satisfy the equation $$ m \frac{d v}{d t}=m g-b v $$ where \( m \) is the mass, \( \boldsymbol{g} \) is the acceleration due to gravity, and \( b>0 \) is a constant (see Figure 2.1). If \( m=100 \mathrm{kg} \), \( g=9.8 \mathrm{m} / \mathrm{sec}^{2}, b=5 \mathrm{kg} / \mathrm{sec} \) and \( v(0)=10 \mathrm{m} / \mathrm{sec} \), solve for \( v(t) \). What is the limiting (i.e., terminal) velocity of the object?

Consider the initial value problem $$ \frac{d y}{d x}+\sqrt{1+\sin ^{2} x} y=x, \quad y(0)=2 $$. (a) Using definite integration, show that the integrating factor for the differential equation can be written as $$ \mu(x)=\exp \left(\int_{0}^{x} \sqrt{1+\sin ^{2} t} d t\right) $$ and that the solution to the initial value problem is $$ y(x)=\frac{1}{\mu(x)} \int_{0}^{x} \mu(s) s d s+\frac{2}{\mu(x)} $$. (b) Obtain an approximation to the solution at \( x=1 \) by using numerical integration (such as Simpson's rule, Appendix C) in a nested loop to estimate values of \( \mu(x) \) and, thereby, the value of $$ \int_{0}^{1} \mu(s) s d s $$. [Hint: First, use Simpson's rule to approximate \( \mu(x) \) at \( x=0.1,0.2, \dots, 1 \). Then use these values and apply Simpson's rule again to approximate \( \int_{0}^{1} \mu(s) s d s . ] \) (c) Use Euler's method (Section 1.4) to approximate the solution at \( x=1 \), with step sizes \( h=0.1 \) and 0.05. [A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]

Newton's Law of Cooling. According to Newton's law of cooling, if an object at temperature \( T \) is immersed in a medium having the constant temperature \( M \), then the rate of change of \( T \) is proportional to the difference of temperature \( M-T \). This gives the differential equation $$ d T / d t=k(M-T) $$ (a) Solve the differential equation for \( T \) (b) A thermometer reading \( 100^{\circ} \mathrm{F} \) is placed in a medium having a constant temperature of \( 70^{\circ} \mathrm{F} \). After 6 min, the thermometer reads \( 80^{\circ} \mathrm{F} \). What is the reading after 20 min? (Further applications of Newton's law of cooling appear in Section 3.3.)

Consider the equation $$\left(y^{2}+2 x y\right) d x-x^{2} d y=0$$ (a) Show that this equation is not exact. (b) Show that multiplying both sides of the equation by $$y^{-2}$$ yields a new equation that is exact. (c) Use the solution of the resulting exact equation to solve the original equation. (d) Were any solutions lost in the process?

In Problems \(9-20\) , determine whether the equation is exact. If it is, then solve it. $$\left(2 x+\frac{y}{1+x^{2} y^{2}}\right) d x+\left(\frac{x}{1+x^{2} y^{2}}-2 y\right) d y=0$$
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