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Chapter 8: Problem 35

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The equation \(u^{\prime}(x)=\left(x^{2} u^{7}\right)^{-1}\) is separable. b. The general solution of the separable equation \(y^{\prime}(t)=\frac{t}{y^{7}+10 y^{4}}\) can be expressed explicitly with \(y\) in terms of \(t\) c. The general solution of the equation \(y y^{\prime}(x)=x e^{-y}\) can be found using integration by parts.

Short Answer

Expert verified
In summary: a. The statement is true since the given equation is separable. b. The statement is false since an explicit solution for y(t) cannot be found. c. The statement is false since integration by parts is not necessary as the equation is separable.

Step by step solution

01

Statement a

To determine if the given equation \(u^{\prime}(x)=\left(x^{2} u^{7}\right)^{-1}\) is separable, we want to rewrite the equation as \(u'(x) = fu(x)\cdot g(x)\), where \(f\) is a function of \(x\) only, and \(g\) is a function of \(u\) only. Rewrite the equation as: \(u^{\prime}(x)=\frac{1}{x^{2} u^{7}}\) Indeed, the equation is separable as we have: \(u^{\prime}(x)=\frac{1}{x^{2}}\cdot\frac{1}{u^{7}}\) In this case, \(f(x) = \frac{1}{x^{2}}\) and \(g(u) = \frac{1}{u^{7}}\). Therefore, the statement is true.
02

Statement b

We are given a separable equation \(y^{\prime}(t)=\frac{t}{y^{7}+10y^{4}}\). In order to determine if an explicit solution can be found, we need to solve the equation by separation of variables. And see if we can solve for \(y(t)\) explicitly. Separate the variables: \(y'(t)(y^{7} + 10y^4) = t\) Integral the equation: \(\int y^{7} + 10y^4 dy = \int t dt\) Calculating the integrals, we have: \(\frac{1}{8}y^{8} + \frac{5}{5}y^5 + C = \frac{1}{2}t^2 + K\) It is evident that expressing the solution for \(y(t)\) explicitly is not possible. Thus, statement b is false.
03

Statement c

We are given the equation \(yy^{\prime}(x)=xe^{-y}\). In order to determine if integration by parts is an applicable method, we try to rewrite it in the form \(u\frac{dy}{dx}=v\), with the intention of choosing appropriate functions for integration by parts. However, if we do so, it appears that this equation is separable: \(\frac{dy}{dx}\frac{1}{y}= \frac{1}{x}e^{-y}\) Separate the variables: \(\frac{dy}{y} = e^{-y} \frac{dx}{x}\) We see that it is unnecessary to use integration by parts because the equation is separable. Assuming that integration by parts is needed for a non-separable equation only, statement c is false.

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Most popular questions from this chapter

Explain how the growth rate function can be decreasing while the population function is increasing.

Consider the general first-order linear equation \(y^{\prime}(t)+a(t) y(t)=f(t) .\) This equation can be solved, in principle, by defining the integrating factor \(p(t)=\exp \left(\int a(t) d t\right) .\) Here is how the integrating factor works. Multiply both sides of the equation by \(p\) (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes $$p(t)\left(y^{\prime}(t)+a(t) y(t)\right)=\frac{d}{d t}(p(t) y(t))=p(t) f(t).$$ Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor. $$y^{\prime}(t)+\frac{3}{t} y(t)=1-2 t, \quad y(2)=0$$

Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. $$y y^{\prime}(x)=\frac{2 x}{\left(2+y^{2}\right)^{2}}, y(1)=-1$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the growth rate function for a population model is positive, then the population is increasing. b. The solution of a stirred tank initial value problem always approaches a constant as \(t \rightarrow \infty\) c. In the predator-prey models discussed in this section, if the initial predator population is zero and the initial prey population is positive, then the prey population increases without bound.

Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.
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