Answer to Question #229238 in Chemical Engineering for Lokika

Question

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}

Answer 1

$\left(D^2-3D+2\right)y=e^{3x}\\ \mathrm{A\:linear\:non-homogeneous\:ODE\:with\:constant\:coefficients\:has\:the\:form\:of}\:\left(a_nD^n+...+a_1D+a_0\right)y=g\left(x\right)\\ \mathrm{The\:general\:solution\:to}\:\left(a_nD^n+...+a_1D+a_0\right)y=g\left(x\right)\:\mathrm{can\:be\:written\:as}\\ y=y_h+y_p\\ _h\mathrm{\:is\:the\:solution\:to\:the\:homogeneous\:ODE}\:\left(a_nD^n+...+a_1D+a_0\right)y=0\\ y_p\mathrm{,\:the\:particular\:solution,\:is\:any\:function\:that\:satisfies\:the\:non-homogeneous\:equation}\\ y=c_1e^{2x}+c_2e^x\\ \mathrm{The\:general\:solution\:}y=y_h+y_p\mathrm{\:is:}\\ y=c_1e^{2x}+c_2e^x+\frac{1}{2}e^{3x}$