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Answer:
The general solution of the given differential equation
[tex]y = ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x[/tex]
Step-by-step explanation:
Step(I):-
Given differential equation
y⁴+18y"+81y=0
⇒ (D⁴+18D²+81)y =0
The auxiliary equation
[tex]m^4+18m^2+81 =0[/tex]
[tex](m^2)^{2} + 2 (9) m^{2} +(9)^2 = 0[/tex]
we will use formula ( a + b)² = a² + 2 a b + b²
⇒ ( m² + 9 ) ² = 0
⇒ ( m² + 9 ) ( m² + 9 ) = 0
[tex]m^{2} =-9\\m= - 3i and m=3i[/tex]
m² + 9 = 0
[tex]m² = -9\\m= -3i and m=3i[/tex]
The complex roots are 0± 3 i ,0 ± 3 i
Step(ii):-
The complementary function
[tex]y = e^{\alpha x } ( c_{1} + c_{2} x ) cos\beta x + ( c_{3} +c_{4} x) sin\beta x[/tex]
The general solution of the given differential equation
[tex]y = e^{0 x } ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x[/tex]
The general solution of the given differential equation
[tex]y = ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x[/tex]
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