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

the opening of this porabola is facing up

step-by-step explanation:

let f (x ) is a function and c be a positive real number,   the graph of the function g (x ) = f (cx ) is represented as follows.

when, c > 1,   the function compresses it in the x - direction.

when, 0 < c < 1, the function stretches it.

the function is g ( x ) = (9x )2.

the heighest power of x in g ( x ) = (9x )2 is 2, so the parent function is square function.

the equation of parent function is f ( x ) = x 2 .

the function is g ( x ) = (9x )2 and the parent function is f ( x ) = x 2 .

so, the functions f and g have the following relationship.

g ( x ) = f ( 9x ).

compare the function g ( x ) = f ( 9x ) with g (x ) = f (cx ).

c = 9 ( > 1 ).

so, the function g (x ) = (9x )2 compresses it in the x - direction.

step-by-step explanation: