Question

Find the functiony_1oftwhich is the solution of

9y'' - 49y = 0 with initial conditions\quad y_1(0) = 1, \quad y_1'(0) = 0 .
y_1 = ? Find the functiony_2oftwhich is the solution of 9y'' - 49y = 0 with initial conditions\quad y_2(0) = 0, \quad y_2'(0) = 1 .
y_2 =? Find the Wronskian W(t) = W(y_1,y_2).

Answer 1

The functions η1(t) and η2(t) are the solutions to the differential equation with distinct initial conditions. Applying these conditions to the general solution provides the specific forms of y1(t) and y2(t), and the Wronskian confirms their linear independence.

Step-by-step explanation:

We need to find the functions η1(t) and η2(t) which are the solutions to the differential equation 9y'' - 49y = 0, including their respective initial conditions. This is a second-order linear homogeneous differential equation with constant coefficients.

Firstly, we can write the characteristic equation:

λ² - (49/9)λ = 0

Solving for λ gives the roots λ = 7/3 and λ = -7/3. Therefore, the general solution to the differential equation is y(t) = C1e^(7/3)t + C2e^(-7/3)t.

The initial conditions for y1(t) are y1(0) = 1, y1'(0) = 0, which when applied to the general solution give us the specific solution for y1(t). In a similar manner, we use the initial conditions for y2(t) are y2(0) = 0, y2'(0) = 1, to find the specific solution for y2(t).

The Wronskian W(t) is a determinant used to check the linear independence of the solutions which in this case will be a non-zero value confirming that y1(t) and y2(t) are linearly independent.