Second Order Linear Constant Coefficents

Routine Name: solcc

Author: Philip Nelson

Language: C++

Description

solcc computes the solution to the second-order linear constant-coefficient equation below at time \(t\).

\[ ay^{\prime \prime} + by^{\prime} + cy = f(t) \]

$ make
$ ./solcc.out

This will compile and run the driver program.

Input

T solcc(T y0, T v0, T a, T b, T c, T t) requires:

T y0 - initial condition \(y(0)\)
T v0 - initial condition \(y^{\prime}(0)\)
T a - coefficent \(a\) on \(y^{\prime\prime}\)
T b - coefficent \(b\) on \(y^{\prime}\)
T c - coefficent \(c\) on \(y\)
T t - time

Note: all parameters must be the same type.

Output

solcc returns a std::optional<std::complex<N>> with the solution.

Code

template <typename T>
T solcc(T y0, T v0, T a, T b, T c, T t)
{
  // roots from the quadratic formula
  std::complex<T> const sqDiscrim = sqrt((b * b) - (4.0 * a * c));
  auto const r1 = (-b + sqDiscrim) / (2.0 * a);
  auto const r2 = (-b - sqDiscrim) / (2.0 * a);

  if (r1 == r2) // double roots
  {
    // calculate c1 and c2
    auto const c1 = y0;
    auto const c2 = v0 - r1 * y0;

    // return the solution
    return std::real(c1 * exp(r1 * t) + c2 * t * exp(r2 * t));
  }
  else // unique roots
  {
    // calculate c1 and c2
    auto const c1 = (v0 - (r2 * y0)) / (r1 - r2);
    auto const c2 = ((r1 * y0) - v0) / (r1 - r2);

    // return the solution
    return std::real(c1 * exp(r1 * t) + c2 * exp(r2 * t));
  }
}

Example

int main()
{
  auto solution = solcc(2.0, 0.0, 3.0, 5.0, -1.0, 3.0);
  std::cout << solution << std::endl;
}

Result

3.13158

Last Modification date: 22 January 2018