Q16E

Question

Find a general solution to the givenhomogeneous equation. ${(D + 1)}^{2} {(D - 6)}^{3} (D + 5) (D^{2} + 1) {(D^{2} + 4)}^{2} [y] = 0$

Step-by-Step Solution

Verified

Answer

The general solution to the homogeneous equation is:

$y = C_{1} e^{- x} + C_{2} x e^{- x} + C_{3} e^{6 x} + C_{4} x e^{6 x} + C_{5} x^{2} e^{6 x} + C_{6} e^{- 5 x} + C_{7} \cos x + C_{8} \sin x + C_{9} \cos 2 x + C_{10} \sin 2 x$

1Step 1: Homogenous Equation

A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. When a row operation is applied to a homogeneous system, the new system is still homogeneous.

2Step 2: Solving of Homogenous Equation :

The given differential equation is ${(D - 1)}^{2} (D - 6) (D + 5) (D^{2} + 1) (D^{2} + 4) [y] = 0$ . To solve this equation we look at its auxillary equation which is . ${(m + 1)}^{2} {(m - 6)}^{3} (m + 5) (m^{2} + 1) (m^{2} + 4) = 0$

3Step 3: Solving for general equation:

The complete set of solution of auxillary equation is ${1 -, - 1, 6, 66, - 5, i, - i, 2 i, - 2 i}$

To conclude that the general solution of the given differential equation is $y = C_{1} e^{- x} + C_{2} x e^{- x} + C_{3} e^{6 x} + C_{4} x e^{6 x} + C_{5} x^{2} e^{6 x} + C_{6} e^{- 5 x} + C_{7} \cos x + C_{8} \sin x + C_{9} \cos 2 x + C_{10} \sin 2 x y = C_{1} e^{- x} + C_{2} x e^{- x} + C_{3} e^{6 x} + C_{4} x e^{6 x} + C_{5} x^{2} e^{6 x} + C_{6} e^{- 5 x} + C_{7} \cos x + C_{8} \sin x + C_{9} \cos 2 x + C_{10} \sin 2 x$ , where $C_{i} (1 \leq i \leq 10)$ are arbitrary constants.

The general solution of the given differential equation is $y = C_{1} e^{- x} + C_{2} x e^{- x} + C_{3} e^{6 x} + C_{4} x e^{6 x} + C_{5} x^{2} e^{6 x} + C_{6} e^{- 5 x} + C_{7} \cos x + C_{8} \sin x + C_{9} \cos 2 x + C_{10} \sin 2 x$ , where $C_{i} (1 \leq i \leq 7)$ are arbitrary constant.

Hence, the final answer is:

$y = C_{1} e^{- x} + C_{2} x e^{- x} + C_{3} e^{6 x} + C_{4} x e^{6 x} + C_{5} x^{2} e^{6 x} + C_{6} e^{- 5 x} + C_{7} \cos x + C_{8} \sin x + C_{9} \cos 2 x + C_{10} \sin 2 x$

Q15E

Q17E

Other exercises in this chapter

Q14E

Find a general solution for the differential equation with x as the independent variable: y(4)+2y'''+10y''+18y'+9y=0

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Q15E

Find a general solution to the givenhomogeneous equation. (D−1)2(D+3)(D2+2D+5)2[y]=0

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Q17E

Find a general solution to the givenhomogeneous equation.(D+4)(D−3)(D+2)3(D2+4D+5)2D5[y]=0

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Q18E

Find a general solution to the givenhomogeneous equation. (D−1)3(D−2)(D2+D+1)(D2+6D+10)3[y]=0

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