Question

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

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

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.

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.

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.