Question

Como encontrar a solução da seguinte EDO-1

dy/dt-6y=1, Y(0)=1

Answer 1

Bom dia

dy/dt - 6y = 1

y(t) = c*e^(6 t) - 1/6
y(0) = 1
c - 1/6 = 1¨
c = 1 + 1/6 = 7/6

y(t) = (7e^(6 t) - 1)/6

Answer 2

$\mathtt{\dfrac{dy}{dt}-6y=1}\\\mathtt{\dfrac{dy}{dt}=1+6y}\\\mathtt{dy=(1+6y)dt}$

$\mathtt{\dfrac{dy}{1+6y}=dt}$

$\mathtt{\int\dfrac{dy}{1+6y}=\int\,dt}\\\mathtt{\dfrac{1}{6}ln|1+6y|=t+c}\\\mathtt{ln|1+6y|=6t+c}$

$\mathtt{{e}^{ln|1+6y|}={e}^{6t+c}}\\\mathtt{1+6y=k{e}^{6t}}\\\mathtt{y(t)=\dfrac{k{e}^{6t}-1}{6}}$

$\mathtt{y(0)=\dfrac{k{e}^{6.0}-1}{6}}\\\mathtt{1=\dfrac{k-1}{6}}\\\mathtt{k-1=6}\\\mathtt{k=6+1}\\\mathtt{k=7}$

$\huge\boxed{\boxed{\mathtt{y(t)=\dfrac{7{e}^{6t}-1}{6}}}}$