Question

y"+2y'+y=0 para y(0) =1 e y (1)=3 qual o resultado

Answer 1

$\large\boxed{\begin{array}{l}\sf y''+2y'+y=0,y(0)=1,y(1)=3\\\sf y(x)=e^{mx}\\\sf y'(x)=me^{mx}\\\sf y''(x)=m^2e^{mx}\\\sf m^2e^{mx}+2me^{mx}+e^{mx}=0\\\sf e^{mx}(m^2+2m+1)=0\\\sf m^2+2m+1=0\\\sf (m+1)^2=1\\\sf m+1=0\\\sf m=-1\\\sf y_h=c_1e^{-x}+c_2xe^{-x}\\\sf y(0)=c_1e^{0}+c_2\cdot 0e^{0}\\\sf c_1=1\\\sf y(1)=e^{-1}+c_2\cdot 1e^{-1}\\\sf 3=e^{-1}(1+c_2)\\\sf 1+c_2=\dfrac{3}{e^{-1}}\\\\\sf 1+c_2=3e\\\sf c_2=3e-1\\\sf y(x)= e^{-x}+(3e-1)xe^{-x}\end{array}}$