A interactive calculator to solve second order differential equations , with constant coefficients, is presented.
A linear second-order homogeneous differential equation with constant coefficients \( a \), \( b \), and \( c \) has the general form [1] , [2] , [3] :
\[
a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + c y = 0
\]
To solve this differential equation using the auxiliary equation (or characteristic equation), we first find the roots of the auxiliary equation , which is obtained by assuming a solution of the form \( y(t) = e^{rt} \), hence \( y'(t) = r e^{rt} \) and \( y''(t) = r^2 e^{rt} \).
Substitute \( y(t) \), hence \( y'(t)\) and \( y''(t) \) in the differential equation and factor as follows
\[
(a r^2 + b r + c) e^{rt} = 0
\]
Since \( e^{rt} \) cannot be equal to zero, we end up with the auxiliary equation corresponding to the differential equation as:
\[
a r^2 + b r + c = 0
\]
1. Write down the auxiliary equation:
\[
a r^2 + b r + c = 0
\]
The nature of the roots of the auxiliary equation determines the behavior of the solutions:
Let \( \Delta = b^2 - 4 \; a \; c \)
1 - If \( \Delta > 0 \) , the roots
\( r_1 = \dfrac{-b+\sqrt{b^2 - 4ac}}{2\;a} \) and \( r_2 = \dfrac{-b-\sqrt{b^2 - 4ac}}{2\;a}\)
are real and distinct. The general solution involves exponential functions as follows.
\[
y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}
\]
where \( C_1 \) and \( C_2 \) are constant to be determined using initial conditions.
2 - If \( \Delta = 0 \) , the roots \( r_1 \) and \( r_2 \) are real and equal to \( -\dfrac{b}{2 \; a} \). The general solution involves a linear function in \( t \) multiplied by an exponential function.
\[
y(t) = ( C_1 + C_2 \; t ) e^{r_1 t}
\]
and \( C_1 \) and \( C_2 \) are constants determined by initial or boundary conditions.
3 - If \( \Delta \lt 0 \) , the roots \( r_1 \) and \( r_2 \) are complex conjugates of the form
\( r_1 = \dfrac{- b + i \sqrt{4 a c - b^2}}{2 \;a} \) and \( r_2 = \dfrac{- b - i \sqrt{4 a c - b^2}}{2 \;a }\)
The general solurion to the differential equation involves sine and cosine functions as follows.
\[
y(t) = e^{\alpha \; t} \left( C_1 \cos(\beta t) + C_2 \sin(\beta \; t) \right)
\]
where
\( \alpha = \dfrac{- b }{2 \;a} \) and \( \beta = \dfrac{ \sqrt{4 a c - b^2} }{2 \;a} \)
and \( C_1 \) and \( C_2 \) are constants determined by initial or boundary conditions.
Enter the coefficients \( a, b , c \) and the initial conditions \( y(0) \) and \( y'(0) \) as real numbers and press "Solve".
Solution