Question

Solve y" + 3y' +2y =1, y(0)=0,y'(0) = 1

Answer 1

$\mathrm{A\:second\:order\:linear,\:non-homogeneous\:ODE\:has\:the\:form\:of\:}\:\:ay''+by'+cy=g\left(x\right)\\ \mathrm{The\:general\:solution\:to\:}a\left(x\right)y''+b\left(x\right)y'+c\left(x\right)y=g\left(x\right)\mathrm{\:can\:be\:written\:as}\\ y=y_h+y_p\\ y_h\mathrm{\:is\:the\:solution\:to\:the\:homogeneous\:ODE\:}a\left(x\right)y''+b\left(x\right)y'+c\left(x\right)y=0\\ y_p\mathrm{,\:the\:particular\:solution,\:is\:any\:function\:that\:satisfies\:the\:non-homogeneous\:equation}\\ y=c_1e^{-t}+c_2e^{-2t}\\ \mathrm{The\:general\:solution\:}y=y_h+y_p\mathrm{\:is:}\\ y=\frac{1}{2}-\frac{1}{2}e^{-2t}$