Math  /  Calculus

QuestionQ(10) Find the general solution of the ODE y(5)y(3)=0y^{(5)}-y^{(3)}=0.

Studdy Solution
Write the general solution based on the roots. For the root r=0 r = 0 with multiplicity 3, the solution is C1+C2x+C3x2 C_1 + C_2 x + C_3 x^2 . For the roots r=1 r = 1 and r=1 r = -1 , the solutions are C4ex C_4 e^x and C5ex C_5 e^{-x} , respectively.
Thus, the general solution is:
y(x)=C1+C2x+C3x2+C4ex+C5ex y(x) = C_1 + C_2 x + C_3 x^2 + C_4 e^x + C_5 e^{-x}
The general solution of the ODE is:
y(x)=C1+C2x+C3x2+C4ex+C5ex y(x) = C_1 + C_2 x + C_3 x^2 + C_4 e^x + C_5 e^{-x}

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