Problems 15 through 24 deal with ODEs and techniques discussed in Section \(4.8 .\) The differential equation $$ \frac{d^{2} x}{d t^{2}}+3 \frac{d x}{d t}+2 x=0 $$ has the general solution $$ x(t)=c_{1} e^{-t}+c_{2} e^{-2 t} \text {. } $$ If we are told that, when \(t=0, x(0)=1\) and its derivative \(x^{\prime}(0)=1\), we can determine \(c_{1}\) and \(c_{2}\) by solving the equations $$ \begin{aligned} &x(0)=c_{1} e^{-0}+c_{2} e^{(-2)}=c_{1}(1)+c_{2}(1)=1, \\ &x^{\prime}(0)=c_{1}(-1) e^{-6}+c_{2}(-2) e^{-0}=-c_{1}-2 c_{2}=1 . \end{aligned} $$ (a) Find the values of \(c_{1}\) and \(c_{2}\) by solving the above. For questions (b) through (e) find the solution of the differential equation subject to the specified initial condition. (b) \(y^{\prime}=10 y ; \quad y(0)=0.001\). (c) \(y^{\prime \prime}-3 y^{\prime}-4 y=0 ; \quad y(0)=0, y^{\prime}(0)=1\). (d) \(y^{\prime \prime}-9 y=0 ; \quad y(0)=5, y^{\prime}(0)=0\). (e) \(y^{\prime \prime}-5 y^{\prime}=0 ; \quad y(0)=1, y^{\prime}(0)=2\).

Initial conditions are essential in solving Ordinary Differential Equations (ODEs) because they help identify specific solutions from a general set of solutions. When dealing with ODEs, we often get a general solution that includes arbitrary constants. Initial conditions allow us to find these constants, transforming the general solution into a specific one. For example, in the given problem, we have the ODE \[ \frac{d^2 x}{dt^2} + 3 \frac{dx}{dt} + 2x = 0 \] and its general solution is \[ x(t) = c_1 e^{-t} + c_2 e^{-2t} \].
The initial conditions provided are:

• \(x(0) = 1\)
• \(x'(0) = 1\)

These initial conditions help us construct a system of linear equations, which we then solve to determine the constants \(c_1\) and \(c_2\). This process turns the general solution into a specific one that satisfies the initial conditions. Using these initial conditions, we can pinpoint the exact behavior of the system at any time \(t\). This makes the solution unique and tailored to the given problem.

Linear equations form the backbone of solving ODEs. In the problem, when we use initial conditions, we set up a system of linear equations. This system can be solved using simple algebraic methods. Here's how it is done:
We start with the general solution of the given ODE: \[ x(t) = c_1 e^{-t} + c_2 e^{-2t} \]
From the initial conditions, we know:

• \(x(0) = c_1 + c_2 = 1\) (since \(e^0 = 1\))
• \(x'(0) = -c_1 - 2c_2 = 1\)

These lead to two linear equations:
\[ c_1 + c_2 = 1 \] \[ -c_1 - 2c_2 = 1 \] Solving these equations simultaneously, we add them to eliminate \(c_1\) and solve for \(c_2\):
\[ (c_1 + c_2) + (-c_1 - 2c_2) = 1 + 1 \ -c_2 = 2 \ c_2 = -2 \] Next, we substitute \(c_2 = -2\) back into the first equation to find \(c_1\): \[ c_1 + (-2) = 1 \ c_1 = 3 \] By solving these linear equations, we find the specific values of \(c_1\) and \(c_2\), which are necessary to fully determine the specific solution of the ODE.

The general solution of an ODE includes arbitrary constants and represents a family of possible solutions. In our example, the general solution for the ODE \[ \frac{d^2 x}{dt^2} + 3 \frac{dx}{dt} + 2x = 0 \] is given by \[ x(t) = c_1 e^{-t} + c_2 e^{-2t} \].
Here, \(c_1\) and \(c_2\) are arbitrary constants that can be any number. They serve to represent the infinite set of solutions that satisfy the differential equation. However, to find a specific solution that fits particular criteria or initial conditions, we must determine these constants. For instance, given initial conditions \(x(0) = 1\) and \(x'(0) = 1\), we derive a specific set of these constants \(c_1\) and \(c_2\) by solving the system of linear equations: \[ c_1 + c_2 = 1\] \[ -c_1 - 2 c_2 = 1 \] Through the methods of solving linear equations, we find \(c_1 = 3\) and \(c_2 = -2\).
Substituting these values back into the general solution gives: \[ x(t) = 3 e^{-t} - 2 e^{-2t} \] This specific solution meets both the differential equation and the initial conditions. Therefore, the general solution transforms into a particular solution that completely defines the system’s behavior under the given conditions.