\[ y^{\prime\prime}(t) + p y^{\prime}(t) + q y(t) = g(t) \]
The constants $p, q$ and the function $g(t)$ are called coefficients. The coefficient $g(t)$ is called the forcing term. If $g(t)=0$, then the equation is said to be homogeneous:
\[ y^{\prime\prime}(t) + p y^{\prime}(t) + q y(t) = 0 \]
The Wronskian of functions $u(t)$ and $v(t)$ is the following function:
$ W(f,g)(t) = u(t) v^{\prime}(t) - u^{\prime}(t) v(t) $
If the Wronskian is non-zero at some point in an interval, then the functions $u(t)$ and $v(t)$ are linearly independent on the interval.
Two functions $y_1(t)$ and $y_2(t)$ form a fundamental set of solutions of a homogeneous equation
\[ y^{\prime\prime}(t) + p y^{\prime}(t) + q y(t) = 0 \]
$y_1(t)$ and $y_2(t)$ are solutions of the homogeneous equation $y^{\prime\prime} + p y^{\prime} + q y = 0$.
The Wronskian $W(y_1,y_2)(t) = y_1(t)y_2^{\prime}(t) - y_1^{\prime}(t) y_2(t)$ is nonzero.
When we are solving a homogeneous equation
\[ y^{\prime\prime}(t) + p y^{\prime}(t) + q y(t) = 0, \] we first find a fundamental set of solutions $y_1(t)$ and $y_2(t)$. Then all the solutions are given as linear combinations of $y_1(t)$ and $y_2(t)$, that is as \[ C_1\, y_1(t) + C_2\, y_2(t). \] This formula is the general solution of the homogeneous equation. Here $C_1$ and $C_2$ are arbitrary constants.Place the cursor over the image to start the animation.
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