What is the solution for 49y^{''}-16y=0,y(0)=1,y^'(0)=0 ?

The solution for 49y^{''}-16y=0,y(0)=1,y^'(0)=0 is y=(1-1/2)e^{(4t)/7}+1/2 e^{-(4t)/7}

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49y^{''}-16y=0,y(0)=1,y^'(0)=0

49 y^{^{''}} - 16 y = 0, y (0) = 1, y^{^{'}} (0) = 0

y = (1 - \frac{1}{2}) e^{\frac{4 t}{7}} + \frac{1}{2} e^{- \frac{4 t}{7}}

Solution steps

49 y^{''} (t) - 16 y = 0

Solve linear ODE:

y = (1 - \frac{1}{2}) e^{\frac{4 t}{7}} + \frac{1}{2} e^{- \frac{4 t}{7}}

y = (1 - \frac{1}{2}) e^{\frac{4 t}{7}} + \frac{1}{2} e^{- \frac{4 t}{7}}

x^{^{''}} + 4 x^{^{'}} + 13 x = 0

y^{^{''}} - 5 y^{^{''}} + 4 y = 0

y^{^{''}} + y^{^{'}} - 2 y = 0, y (0) = 0, y^{^{'}} (0) = 0

y^{^{'''}} + 5 y^{^{''}} - 25 y^{^{'}} - 125 y = 0

35 y^{^{''''}} + 237 y^{^{''}} + 400 y = 0

What is the solution for 49y^{''}-16y=0,y(0)=1,y^'(0)=0 ?
The solution for 49y^{''}-16y=0,y(0)=1,y^'(0)=0 is y=(1-1/2)e^{(4t)/7}+1/2 e^{-(4t)/7}