which of the following can help astronomers detect if a star is part of a binary star system A. the star varies temperatures near the binary star B. the star wobbles near the binary star C. the star has very strong gravity D. the
Physics
A planet has two satellite moons. Moon X has an orbital period of 2.13 days. Moon Y has an orbital period of about 3.53 days. Both moons have nearly circular orbits. Use Kepler's third law to find the distance of each satellite
Physics 1
In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun (Msun = 1.99 × 10^(30) kg) at a distance of 1.50 ×
Physics
planet orbits a star once every 2.84 10^7 s in a nearly circular orbit of radius 1.42 10^11 m. (a) With respect to the star, determine the angular speed of the planet. (b) With respect to the star, determine the tangential speed
Flex: 0 auto; This is the same as flex: initial; and the shorthand for the default value: flex: 0 1 auto. It sizes the item based on its width/height properties (or its content if not set). It makes the flex item inflexible when there is some free space left, but allows it to shrink to its minimum when there is not enough space. For such a projectile with an initial vertical velocity of 39.2 m/s, it would take 4 seconds for it to reach the peak where its vertical velocity is 0 m/s. With this notion in mind, it is evident that the time for a projectile to rise to its peak is a matter of dividing the vertical component of the initial velocity (v iy ) by the acceleration. Windows 1.0 is a graphical personal computer operating environment developed by Microsoft.Microsoft had worked with Apple Computer to develop applications for Apple's January 1984 original Macintosh, the first mass-produced personal computer with a graphical user interface (GUI) that enabled users to see user friendly icons on screen. Windows 1.0 was released on November 20, 1985, as the first.
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What is the solution of the Homogeneous Differential Equation? : #dy/dx = (x^2+y^2-xy)/x^2# with #y(1)=0#
1 Answer
Explanation:
We have:
# dy/dx = (x^2+y^2-xy)/x^2 # with #y(1)=0#
Which is a First Order Nonlinear Ordinary Differential Equation. Let us attempt a substitution of the form:
# y = vx #
Differentiating wrt #x# and applying the product rule, we get:
# dy/dx = v + x(dv)/dx #
Substituting into the initial ODE we get:
# v + x(dv)/dx = (x^2+(vx)^2-x(vx))/x^2 #
Then assuming that #x ne 0# this simplifies to:
# v + x(dv)/dx = 1+v^2-v #
# :. x(dv)/dx = v^2-2v+1 #
And we have reduced the initial ODE to a First Order Separable ODE, so we can collect terms and separate the variables to get:
# int 1/(v^2-2v+1) dv = int 1/x dx #
# int 1/(v-1)^2 dv = int 1/x dx #
Both integrals are standard, so we can integrate to get:
# -1/(v-1) = ln|x| + C #
Using the initial condition, # y(1)=0 => v(1)=0 #, we get:
# -1/(0-1) = ln|1| + C => 1#
Thus we have:
# -1/(v-1) = ln|x| +1 #
# :. 1-v = 1/(1+ln|x|) #
# :. v = 1 - 1/(1+ln|x|) #
# = (1+ln|x|-1)/(1+ln|x|) #
# = (ln|x|)/(1+ln|x|) #
Then, we restore the substitution, to get the General Solution: