Question

A. Find the first and second derivative of the following function in terms of the parameters c1 and c2.

Answer 1

Answer:

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 )

$C_{2} = \frac{y^{11} }{12 x^{2} }$ ...(a)

Step(ii):-

Substitute (a) in equation (ii)

$y^{l} = C_{1} + (\frac{y^{ll} }{12x^{2} } ) (4 x^{3} )$

$C_{1} = y^{l} - (\frac{y^{ll} }{12 x^{2} } ) (4 x^{3} )$ ...(b)

$C_{2} = \frac{y^{11} }{12 x^{2} }$

Step(iii):-

$y = (y^{l} - (\frac{y^{ll} }{12 x^{2} } ) (4 x^{3} ) x + \frac{y^{ll} }{12 x^{2} } x^{4}$

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