Solve the equation y' = x/r? + 1/(ye") and graph sev-eral members of the family of solutions (if your CASdoes implicit plots). How does the solution curve changeas the constant C varies?

the given differential equation is: y'=xx2+1yey we have to graph the several members of the family…

Answered: Solve the equation y' = x/r? + 1/(ye")…

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter2: Graphical And Tabular Analysis

Section2.1: Tables And Trends

Problem 1TU: If a coffee filter is dropped, its velocity after t seconds is given by v(t)=4(10.0003t) feet per...

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Question

Solve the equation y' = x/r? + 1/(ye") and graph sev-
eral members of the family of solutions (if your CAS
does implicit plots). How does the solution curve change
as the constant C varies?

Expert Solution

Step 1

the given differential equation is:

$y' = \frac{x \sqrt{x^{2} + 1}}{y e^{y}}$

we have to graph the several members of the family of solutions.

then we have to determine how does the solution curve change as the constant C varies.

the given differential equation can be written as:

$\begin{array}{rcl} y' & = & \frac{x \sqrt{x^{2} + 1}}{y e^{y}} \\ \frac{d y}{d x} & = & \frac{x \sqrt{x^{2} + 1}}{y e^{y}} \\ y e^{y} d y & = & x \sqrt{x^{2} + 1} d x (1) \end{array}$

Step 2

now integrate the both sides of the equation (1),

$\int y e^{y} d y = \int x \sqrt{x^{2} + 1} d x (2)$

let the integral $\int y e^{y} d y$ be $I_{1}$

therefore,

$I_{1} = \int y e^{y} d y$

integrate the integral $I_{1}$ by using the integration by parts.

therefore,

$\begin{array}{rcl} I_{1} & = & \int y e^{y} d y \\ = & y \int e^{y} d y - \int (\frac{d y}{d y} \int e^{y} d y) d y \\ = & y (e^{y}) - \int 1 \times e^{y} d y \\ = & y e^{y} - e^{y} \end{array}$

Step 3

now let the integral $\int x \sqrt{x^{2} + 1} d x$ be $I_{2}$

therefore,

$I_{2} = \int x \sqrt{x^{2} + 1} d x$

let $x^{2} + 1 = t$

therefore,

$\begin{array}{rcl} d (x^{2} + 1) & = & d t \\ 2 x d x & = & d t \\ x d x & = & \frac{d t}{2} \end{array}$

now substitute these values in the integral $I_{2}$

Step 4

therefore,

$\begin{array}{rcl} I_{2} & = & \int x \sqrt{x^{2} + 1} d x \\ = & \int \sqrt{t} \frac{d t}{2} \\ = & \frac{1}{2} \int t^{\frac{1}{2}} d t \\ = & \frac{1}{2} [\frac{t^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}] \\ = & \frac{1}{2} \times \frac{t^{\frac{3}{2}}}{\frac{3}{2}} \\ = & \frac{1}{2} \times \frac{2}{3} t^{\frac{3}{2}} \\ = & \frac{1}{3} t^{\frac{3}{2}} \end{array}$

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Solve the equation y' = x/r? + 1/(ye") and graph sev- eral members of the family of solutions (if your CAS does implicit plots). How does the solution curve change as the constant C varies?