# Obtain the Differential equations: parabolas with vertex and focus on the x-axis.

Question
Differential equations
Obtain the Differential equations: parabolas with vertex and focus on the x-axis.

2020-11-11
Let M denote the family of parabolas whose vertex and focus both on the x-axis and let (a,0) be the focus of a member of the given family, where a is an arbitrary constant. Therefore, equation of family M is $$\displaystyle{y}^{{2}}={4}{a}{x}$$...(1) Differentiating both sides of equation with respect to x, we get
$$\displaystyle{2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={4}{a}{x}$$
Substituting the value of 4a from equation in (1) $$\displaystyle{y}^{{2}}={\left({2}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}\right)}{x}$$
$$\displaystyle\to{y}^{{2}}={2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}$$
$$\displaystyle\to{y}^{{2}}-{2}{x}{y}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}=-$$
which is the Differential equations of the family of parabolas.

### Relevant Questions

Find the vertex, focus, and directrix for the parabolas:
a) $$\displaystyle{\left({y}–{9}\right)}^{{2}}={8}{\left({x}-{2}\right)}$$
b) $$\displaystyle{y}^{{2}}–{4}{y}={4}{x}–{2}^{{2}}$$
c) $$\displaystyle{\left({x}–{6}\right)}^{{2}}={4}{\left({y}–{2}\right)}$$
For Exercise, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is. If the equation represents a circle, identify the center and radius. • If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. • If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. • If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. $$x2\+\ y2\ −\ 4x\ −\ 6y\ +\ 1 = 0$$
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3,3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2,4) and focus at (-4, 4)
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3, 3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2, 4) and focus at (-4, 4)
a) determine the type of conic b) find the standard form of the equation Parabolas: vertex, focus, directrix Circles: Center, radius Ellipses: center, vertices, co-vertices, foci Hyperbolas: center, vertices, co-vertices, foci, asymptotes $$16x^2 + 64x - 9y^2 + 18y - 89 = 0$$
Solve the differential equations
(1) $$\displaystyle{x}{y}'-{2}{y}={x}^{{3}}{e}^{{x}}$$
(2) $$\displaystyle{\left({2}{y}{\left.{d}{x}\right.}+{\left.{d}{y}\right.}\right)}{e}^{{2}}{x}={0}$$
Solve the Differential equations $$y'' + 4y = 4 \tan^{2} x$$
Find dw/dt using the appropriate Chain Rule. Evaluate $$\frac{dw}{dt}$$ at the given value of t. Function: $$w=x\sin y,\ x=e^t,\ y=\pi-t$$ Value: t = 0
The coefficient matrix for a system of linear differential equations of the form $$\displaystyle{y}^{{{1}}}={A}_{{{y}}}$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system.
$$\displaystyle{\left[\lambda_{{{1}}}=-{1}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}{0}{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{{2}}}={3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}-{i}{1}+{i}{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{3}}=-{3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}+{i}{1}-{i}-{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right]}$$
$$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={a}{y}+{b}{y}^{{2}}$$