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 )
[tex]C_{2} = \frac{y^{11} }{12 x^{2} }[/tex] ...(a)
Step(ii):-
Substitute (a) in equation (ii)
[tex]y^{l} = C_{1} + (\frac{y^{ll} }{12x^{2} } ) (4 x^{3} )[/tex]
[tex]C_{1} = y^{l} - (\frac{y^{ll} }{12 x^{2} } ) (4 x^{3} )[/tex] ...(b)
[tex]C_{2} = \frac{y^{11} }{12 x^{2} }[/tex]
Step(iii):-
[tex]y = (y^{l} - (\frac{y^{ll} }{12 x^{2} } ) (4 x^{3} ) x + \frac{y^{ll} }{12 x^{2} } x^{4}[/tex]
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
