Question

Find the solution to the boundary value problem: d2ydt2−7dydt+10y=0, y(0)=6,y(1)=3

Answer 1

Try this solution. Basic steps marked by '·'
You may simplify the answer.

Answer 2

The solution to the boundary value problem: d²y/dt² - 7dy/dt + 10y = 0, y(0)=6,y(1)=3 is y = {(6$e^{5}$ - 3)/($e^{5}$ - $e^{2}$)}$e^{2x}$ + {(3 - 6)/($e^{5}$  - $e^{2}$)}  $e^{5x}$.

What is the boundary value problem?

A boundary value problem represents a system of ordinary differential equations that has solution and derivative values mentioned at multiple points.

d²y/dt² - 7dy/dt + 10y = 0

Let, d²/dt² = D², d/dt = D.

Then, D²y - 7Dy + 10y =  0

⇒ y(D² -7D + 10) = 0

⇒ (D² -7D + 10) = 0

⇒ (D² -5D - 2D + 10) = 0

⇒ D(D - 5) - 2(D - 5) = 0

⇒ (D - 5)(D - 2) = 0

⇒ D = 2, 5

Therefore, y = C₁ $e^{(D1) x}$ + C₂ $e^{(D2)x}$

y = C₁ $e^{2x}$ + C₂$e^{5x}$

Putting the values of y, we get:

C₁ $e^{0}$ + C₂$e^{0}$ = 6

C₁ + C₂ = 6 ------(1)

Similarly,

C₁ $e^{2}$ + C₂$e^{5}$ = 3

⇒ C₁ + C₂$e^{3}$ = 3/$e^{2}$-----(2)

Solving equation (1) and (2), we get:

C₁ = [6$e^{5}$ - 3]/[$e^{5}$ - $e^{2}$]

C₂ = [3 - 6$e^{2}$]/[$e^{5}$  - $e^{2}$]

Hence, y = {(6$e^{5}$ - 3)/($e^{5}$ - $e^{2}$)}$e^{2x}$ + {(3 - 6)/($e^{5}$  - $e^{2}$)}  $e^{5x}$

Learn more about boundary value problems here: https://brainly.com/question/16735055

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