Question

3 d^ 2 y dx^ 2 + dy dx -14y=0;y(0)=1,y^ prime (0)= -1

Answer 1

$3y''+y'-14=0\\ \textsf{The auxiliary equation is}\\ 3m²+m-14\\ m_1=2, m_2=\frac{-7}{3}\\ \textsf{Hence, the general solution is}\\ y=Ae^{2x}+Be^{\frac{-7}{3}x}\\ \therefore y'=2Ae^{2x}-\frac73Be^{\frac{-7}{3}x}\\ \textsf{Given,}\\ y(0)=1, y'(0)=-1\\ \textsf{substituting the given conditions, we have}\\ 1=A+B\\ B=1-A\\ \textsf{Also,}\\ -1=2A-\frac73B\\ -3=6A-7+7A\\ 13A=4\\ A=\frac4{13},\\ B=\frac9{13}\\ \textsf{Hence, the particular solution is}\\ y=\frac4{13}e^{2x}+\frac9{13}e^{-\frac73x}\\$