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Answered question

2022-03-24

Determine a unique solution of the separable differential equa- tion that satisfies the given initial condition.

\frac{d^{2} y}{{d t}^{2}} = \frac{d y}{d t} \frac{d y}{d t} (0) = 2, y (0) = 3

pypberissootcu

Beginner2022-03-25Added 14 answers

Given Differential equation is

\frac{d^{2} y}{{d t}^{2}} = \frac{d y}{d t}

Boundary Condition is

\frac{d y}{d t} (0) = 2

and

y (0) = 3

Solution:

\frac{d^{2} y}{{d t}^{2}} = \frac{d y}{d t}

Substitute

\frac{d y}{d t} = y^{'}

So,

y^{″} = y^{'}

y^{″} - y^{'} = 0

This is second order linear homogeneous ordinary differential equation
Auxillary equation is

m^{2} - m = 0

m (m - 1) = 0

Roots are

m = 0

and

m = 1

Roots are real and district
So solution is

y (t) = C_{1} e^{m_{1} t} + C_{2} e^{m_{2} t}

Here put

m_{1} = 0

and

m_{2} = 1

So solution is

y = C_{1} e^{0 \cdot t} + C_{2} e^{1 \cdot t}

y (t) = C_{1} + C_{2} e^{t}

Use Given Initial Condition

y (0) = C_{1} + C_{2} e^{0} = C_{1} + C_{2} = 3

y (t) = C_{1} + C_{2} e^{T}

y^{'} (t) = C_{2} e^{t}

y 0) = C_{2} = 2

\Rightarrow C_{1} + 2 = 3

C_{1} = 3 - 2 = 1

\Rightarrow C_{1} = 1

Put

C_{1} = 1

and

C_{2}

in y(t)

y (t) = 1 + 2 e^{t}

y (t) = 1 + 2 e^{t}

is solution of given Differential Equation

Jeffrey Jordon

Expert2022-03-31Added 2605 answers

Answer is given below (on video)

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