91Ó°ÊÓ

The initial condition in solving differential equations is crucial. It helps us find the unique solution to a given problem. In the provided exercise, the initial condition is given as $$ y(0)=1 $$. This means at time t=0, the value of the function y(t) is 1.

Without an initial condition, we can only determine the general solution of a differential equation, like $$ y(t)=Ce^{3t} $$, which contains an arbitrary constant \( C \).

To find the particular solution, we need to use the initial condition to find the specific value of \( C \). In this problem, we substitute $$ t=0 $$ and $$ y(0)=1 $$ into the general solution, giving us: $$ 1=Ce^{3(0)}=C $$.

Therefore, the value of \( C \) is 1. This makes the particular solution specific to our initial condition.

The exponential function is key in solving many differential equations, notably when dealing with equations of the form \( y^{\text{'}}=ky \). In our particular example, the solution involves the exponential function, $$ y(t)=e^{kt} $$.

Here is why:

• The general solution to \( y^{\text{'}}=3y \) is expressed as \( y(t)=Ce^{3t} \). The exponential function \( e^{3t} \) indicates rapid growth due to the exponent being multiplied by time \( t \).


• The initial condition helps determine the constant \( C \) for the exponential, ensuring an accurate particular solution.


• The base of the exponential function, \( e \), is a mathematical constant approximately equal to 2.71828. It uniquely retains the property where the rate of growth (the derivative) is proportional to the function itself.

Once we have determined the general solution of the differential equation, we use the initial conditions to find a unique particular solution. For the given differential equation \( y^{\text{'}}=3y \) with initial condition \( y(0)=1 \), the steps are as follows:

• First, write the general solution as \( y(t)=Ce^{3t} \).


• Next, apply the initial condition \( y(0)=1 \).


• By substituting \( t=0 \) and \( y(0)=1 \), solve for practically the constant \( C \): 1=\( Ce^{3(0)}=C \), so \( C=1 \).

Thus, the particular solution is \( y(t)=e^{3t} \), aligning uniquely with the initial condition. This particular solution tells us the specific behavior of the system over time, given the starting point \( y(0)=1 \).

In summary, the particular solution captures how the system evolves specifically from that provided initial state.