Solution of (D³ - 2D² - 3D)y = 0
A) y = C1.₁e^{x} + C₂е^{-x }+ С3.e^{3x} B) y = C1e^x + C₂е^{-x} + С3.e^{-3x}
C) y = C1+C₂е^{-x} + С3.e^{3x}
D) y = C₁+C₂е^{-x} + С3e^{-3x}
(D3−2D2−3D)y=0A linear homogeneous ODE with constant coefficients has the form of (anDn+...+a1D+a0)y=0D=0, D=−1, D=3For non repeated real roots D1, D2, ..., Dn, the general solution takes the form:y=c1eD1 x+c2eD2 x+...+cneDn xc1e0+c2e−x+c3e3xy=c1+c2e−x+c3e3x\left(D^3-2D^2-3D\right)y=0\\ \mathrm{A\:linear\:homogeneous\:ODE\:with\:constant\:coefficients\:has\:the\:form\:of}\:\left(a_nD^n+...+a_1D+a_0\right)y=0\\ D=0,\:D=-1,\:D=3\\ \mathrm{For\:non\:repeated\:real\:roots\:}D_1,\:D_2,\:...,\:D_n\mathrm{,\:the\:general\:solution\:takes\:the\:form:}\\ y=c_1e^{D_1\:x}+c_2e^{D_2\:x}+...+c_ne^{D_n\:x}\\ c_1e^0+c_2e^{-x}+c_3e^{3x}\\ y=c_1+c_2e^{-x}+c_3e^{3x}(D3−2D2−3D)y=0AlinearhomogeneousODEwithconstantcoefficientshastheformof(anDn+...+a1D+a0)y=0D=0,D=−1,D=3FornonrepeatedrealrootsD1,D2,...,Dn,thegeneralsolutiontakestheform:y=c1eD1x+c2eD2x+...+cneDnxc1e0+c2e−x+c3e3xy=c1+c2e−x+c3e3x
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