Chapter 2: Problem 18
Show that any separable equation, $$ M(x)+N(y) y^{\prime}=0 $$ is also exact.
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chapter 2: Problem 18
Show that any separable equation, $$ M(x)+N(y) y^{\prime}=0 $$ is also exact.
These are the key concepts you need to understand to accurately answer the question.
All the tools & learning materials you need for study success - in one app.
Get started for freeDetermine whether or not each of the equations is exact. If it is exact, find the solution. $$ (2 x+3)+(2 y-2) y^{\prime}=0 $$
Consider the sequence \(\phi_{n}(x)=2 n x e^{-n x^{2}}, 0 \leq x \leq 1\) (a) Show that \(\lim _{n \rightarrow \infty} \phi_{n}(x)=0\) for \(0 \leq x \leq 1\); hence $$ \int_{0}^{1} \lim _{n \rightarrow \infty} \phi_{n}(x) d x=0 $$ (b) Show that \(\int_{0}^{1} 2 n x e^{-x x^{2}} d x=1-e^{-x}\); hence $$ \lim _{n \rightarrow \infty} \int_{0}^{1} \phi_{n}(x) d x=1 $$ Thus, in this example, $$ \lim _{n \rightarrow \infty} \int_{a}^{b} \phi_{n}(x) d x \neq \int_{a}^{b} \lim _{n \rightarrow \infty} \phi_{n}(x) d x $$ even though \(\lim _{n \rightarrow \infty} \phi_{n}(x)\) exists and is continuous.
solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value \(y_{0}\). $$ y^{\prime}=2 t y^{2}, \quad y(0)=y_{0} $$
Show that the equations are not exact, but become exact when multiplied by the given integrating factor. Then solve the equations. $$ (x+2) \sin y d x+x \cos y d y=0, \quad \mu(x, y)=x e^{x} $$
Suppose that a certain population obeys the logistic equation \(d y / d t=r y[1-(y / K)]\). (a) If \(y_{0}=K / 3\), find the time \(\tau\) at which the initial population has doubled. Find the value of \(\tau\) corresponding to \(r=0.025\) per year. (b) If \(y_{0} / K=\alpha,\) find the time \(T\) at which \(y(T) / K=\beta,\) where \(0<\alpha, \beta<1 .\) Observe that \(T \rightarrow \infty\) as \(\alpha \rightarrow 0\) or as \(\beta \rightarrow 1 .\) Find the value of \(T\) for \(r=0.025\) per year, \(\alpha=0.1\) and \(\beta=0.9 .\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.