Given equation is y′′′′+2y′′′+11y′′+18y′+18y=0
Making auxiliary equation, m4+2m3+11m2+18m+18=0
Roots of the equation are m=−1±i,±3i
So equation will be y=e−x(c1cosx+c2sinx)+c3cos3x+c4sin3x
applying boundary conditions y(0)=2,y′(0)=3,y′′(0)=−11,y′′′(0)=−23
simply putting x=0 in equation, we obtain
2=c1+c3 (1)
differentiating y once and applying condition , we get
y′=−e−x(c1cos(x)+c2sin(x))+e−x(−c1sin(x)+c2cos(x))−3c3sin(3x)+3c4cos(3x)
3=−c1+c2+3c4 (2)
differentiating again and applying conditions
y′′=2c1e−xsin(x)−2c2e−xcos(x)−9c3cos(3x)−9c4sin(3x)
−11=−2c2−9c3 (3)
differentiating again and applying conditions
y′′′=2c1(−e−xsin(x)+e−xcos(x))−2c2(−e−xcos(x)−e−xsin(x))+27c3sin(3x)−27c4cos(3x)
−23=2c1+2c2−27c4 (4)
solving equation 1,2,3 and 4,
c1=c2=c3=c4=1
Then equation will be
y=e−x(cosx+sinx)+cos3x+sin3x
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