Question

Find the function y₁of t which is the solution of 49y ′′+140y ′+75y=0 with initial conditions y₁ (0)=1, y₁′ (0)=0. y₁ = Find the function y₂ of t which is the solution of 49y′′+140y′+75y=0 with initial conditions y₂(0)=0, y'₂(0)=1 y₂= Find the Wronskian W(t)=W(y₁,y₂ ) W(t)= Remark: You can find W by direct computation and use Abel's theorem as a check. You should find that W is not zero and so y₁ and y₂form a fundamental set of solutions of 49y′′+140y′+75y=0.

Answer 1

The solution to the differential equation 49y'' + 140y' + 75y = 0 is found by solving the characteristic equation and applying initial conditions for y1 and y2. The Wronskian W(t) is computed to confirm that y1 and y2 are linearly independent and form a fundamental set of solutions, and Abel's theorem serves as a verification method.

Step-by-step explanation:

To find the functions y1 of t and y2 of t that solve the differential equation 49y'' + 140y' + 75y = 0 with given initial conditions, we follow these steps:

1. Write down the characteristic equation associated with the differential equation, which is r2 + b/m * r + k/m = 0, with m = 49, b = 140, and k = 75.
2. Solve the characteristic equation for r to find two solutions r1 and r2.
3. Use the solutions to write the general solution of the differential equation as a combination of exponential functions.
4. Apply the initial conditions to find the constants in the general solution.

For the initial conditions y1(0) = 1 and y1'(0) = 0, we substitute and solve for the constants. Similarly, for the initial conditions y2(0) = 0 and y2'(0) = 1, we again substitute and solve for different constants to find y2.

To find the Wronskian W(t), which is a determinant consisting of y1, y1', y2, and y2', we directly calculate it using the solutions. The Wronskian confirms the solutions are linearly independent and form a fundamental set.

Abel's Theorem Verification

As a verification step, we can use Abel's theorem which states that the Wronskian of two solutions to a second-order linear homogeneous differential equation with continuous coefficients is either always zero or never zero. Since we found a non-zero Wronskian, this indicates that y1 and y2 do indeed form a fundamental set of solutions, which aligns with the theorem.