A. Find the first and second derivative of the following function in terms of the parameters c1 and c2. y = c1x + c2x4 y′

b. Find a linear second-order differential equation F(x, y, y′, y″) = 0 for which y = c1x + c2x4 is a two-parameter family of solutions. Make sure that your equation is free of the arbitrary parameters c1 and c2. (Use yp for y′ and ypp for y″.)

Linear second-order differential equation F(x, y, y′, y″) = 0

y¹¹( -3 x⁴ )+ y¹12 x³ - 12 x² y= 0  

Step-by-step explanation:

Step(i):-

Given differential equation

y = c₁x + c₂x⁴  ...(i)

Differentiating equation (i) with respective to 'x', we get

y¹ = c₁(1) +  c₂ ( 4 x³ ) ...(ii)

Differentiating equation (ii) with respective to 'x', we get

y¹¹ =   c₂ ( 12 x² )

...(a)

Step(ii):-

Substitute (a) in equation (ii)

...(b)

Step(iii):-

12 x² y = (y¹12 x³ - 4 x⁴ y¹¹) + x⁴ y¹¹

x⁴ y¹¹ - 4 x⁴ y¹¹ + y¹12 x³ - 12 x² y= 0

y¹¹( -3 x⁴ )+ y¹12 x³ - 12 x² y= 0

a) one student is chosen from classroom a, then a student is chosen from classroom b.  

step-by-step explanation:

independent events are ones in which the probability of one event does not affect the probability of the second event.

one student is chosen from classroom a; this does not affect the chances of choosing a student from classroom b.   this makes these independent events.

when one card is chosen from a standard deck and then set aside before a second card is chosen, the probability of the second card being chosen changes.   this makes them dependent events.

when one student is chosen from classroom a and then a second student is chosen from classroom a, the probability of choosing the second student changes.   this makes them dependent events.

when a letter is chosen from the alphabet and then a second letter is chosen, the probability of choosing the second letter changes.   this makes them dependent events.

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step-by-step explanation: