k4s3thg368

2022-03-24

Given: ${y}^{2}=c{e}^{x}-x-1$

Find the orthogonal family of curves and write 2 curves from both families that pass in (2,0).

Find the orthogonal family of curves and write 2 curves from both families that pass in (2,0).

Makenzie Hart

Beginner2022-03-25Added 8 answers

Step 1

To find the orthogonal family of curves, you need to first solve for c in terms of x and y.

${y}^{2}=c{e}^{x}-x-1$

$2y\sim {y}^{\prime}=c{e}^{x}-1$

From the given family of curves, $c=\frac{1+x+{y}^{2}}{{e}^{x}}$

So, $y}^{\prime}=\frac{x+{y}^{2}}{2y$

Slope for the orthogonal trajectories will be $\sim {y}^{\prime}=-\frac{2y}{x+{y}^{2}}$

Or, $\sim 2y\sim dx+(x+{y}^{2})\sim dy=0$, which is of the form $M\sim dx+N\sim dy=0$.

Step 2

To solve the above differential equation, note that it is not exact $({M}_{y}\ne {N}_{x})$ so first find an integrating factor that makes it exact, which in this case is $\frac{1}{\sqrt{y}}$.

Multiplying by the integrating factor,

$2\sqrt{y}\sim dx+(\frac{x}{\sqrt{y}}+{y}^{\frac{3}{2}})dy=0$

As $M}_{y}={N}_{x$, it is now exact and integrating, we get the solution

$2x\sqrt{y}+{\frac{25}{y}}^{\frac{5}{2}}=C$

To get two curves from both families that pass through the point (2,0), plug in $x=2,y=0$ in the equation of curves and find values of c and C.

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.