{"type":"rich","version":"1.0","title":"Substituting the exponential trial solution into the homogeneous equation","html":"<span class=\"wb-embed wb-embed-math\" data-latex=\"y=e^{rx},\\quad y'=r\\,e^{rx},\\quad y''=r^{2}\\,e^{rx};\\qquad y''+a\\,y'+b\\,y=\\bigl(r^{2}+a\\,r+b\\bigr)e^{rx}=0\\ \\Rightarrow\\ r^{2}+a\\,r+b=0,\\quad e^{rx}\\neq 0\">y=e^{rx},\\quad y'=r\\,e^{rx},\\quad y''=r^{2}\\,e^{rx};\\qquad y''+a\\,y'+b\\,y=\\bigl(r^{2}+a\\,r+b\\bigr)e^{rx}=0\\ \\Rightarrow\\ r^{2}+a\\,r+b=0,\\quad e^{rx}\\neq 0</span>","width":640,"height":320}