Answer to Question #229238 in Chemical Engineering for Lokika

Question #229238

The yp value of (D²− 3D+ 2)y = e^{3x}, choose the correct answer


A) 1/2 e^{3x}


B) e^{3x}


C) 2e^{3x}. D) 3e^{3x}


1
Expert's answer
2021-09-04T01:00:59-0400

(D23D+2)y=e3xAlinearnonhomogeneousODEwithconstantcoefficientshastheformof(anDn+...+a1D+a0)y=g(x)Thegeneralsolutionto(anDn+...+a1D+a0)y=g(x)canbewrittenasy=yh+yphisthesolutiontothehomogeneousODE(anDn+...+a1D+a0)y=0yp,theparticularsolution,isanyfunctionthatsatisfiesthenonhomogeneousequationy=c1e2x+c2exThegeneralsolutiony=yh+ypis:y=c1e2x+c2ex+12e3x\left(D^2-3D+2\right)y=e^{3x}\\ \mathrm{A\:linear\:non-homogeneous\:ODE\:with\:constant\:coefficients\:has\:the\:form\:of}\:\left(a_nD^n+...+a_1D+a_0\right)y=g\left(x\right)\\ \mathrm{The\:general\:solution\:to}\:\left(a_nD^n+...+a_1D+a_0\right)y=g\left(x\right)\:\mathrm{can\:be\:written\:as}\\ y=y_h+y_p\\ _h\mathrm{\:is\:the\:solution\:to\:the\:homogeneous\:ODE}\:\left(a_nD^n+...+a_1D+a_0\right)y=0\\ y_p\mathrm{,\:the\:particular\:solution,\:is\:any\:function\:that\:satisfies\:the\:non-homogeneous\:equation}\\ y=c_1e^{2x}+c_2e^x\\ \mathrm{The\:general\:solution\:}y=y_h+y_p\mathrm{\:is:}\\ y=c_1e^{2x}+c_2e^x+\frac{1}{2}e^{3x}


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