To find the solution of the given differential equation \(\frac{{dy}}{{dx}} + 4y = 5\), with the initial condition \(y(0) = 2.25\), we will follow a step-by-step process. This is a first-order linear differential equation, which can be solved using the integrating factor method.

Differential Equation Structure

The given differential equation is:

\[\frac{{dy}}{{dx}} + 4y = 5\]

This equation is in the standard form of a first-order linear differential equation, which is:

\[\frac{{dy}}{{dx}} + P(x)y = Q(x)\]

By comparing our equation with the standard form, we can identify \(P(x)\) and \(Q(x)\):

• \(P(x)\): The coefficient of \(y\) is \(4\). So, \(P(x) = 4\).
• \(Q(x)\): The term on the right side of the equation is \(5\). So, \(Q(x) = 5\).

Integrating Factor Calculation

The next step is to calculate the integrating factor (IF). The formula for the integrating factor is:

\[IF = e^{\int P(x) dx}\]

Substitute the value of \(P(x) = 4\) into the formula:

\[IF = e^{\int 4 dx}\]

Integrating \(4\) with respect to \(x\) gives \(4x\):

\[IF = e^{4x}\]

So, the integrating factor for this differential equation is \(e^{4x}\).

General Solution Derivation

Now, multiply the entire differential equation by the integrating factor \(e^{4x}\). This transforms the left side into the derivative of a product:

\[e^{4x} \frac{{dy}}{{dx}} + 4y e^{4x} = 5 e^{4x}\]

The left side, \(e^{4x} \frac{{dy}}{{dx}} + 4y e^{4x}\), is exactly the result of applying the product rule to \(\frac{d}{dx}(y \cdot e^{4x})\). Therefore, we can rewrite the equation as:

\[\frac{d}{dx}(y e^{4x}) = 5 e^{4x}\]

To find the general solution \(y\), integrate both sides of the equation with respect to \(x\):

\[\int \frac{d}{dx}(y e^{4x}) dx = \int 5 e^{4x} dx\]

On the left side, the integral cancels out the derivative:

\[y e^{4x} = 5 \int e^{4x} dx\]

To integrate \(e^{4x}\), we use a simple substitution (or recall the rule \(\int e^{ax} dx = \frac{1}{a} e^{ax}\)):

\[y e^{4x} = 5 \left(\frac{e^{4x}}{4}\right) + C\]

Here, \(C\) is the constant of integration. So we have:

\[y e^{4x} = \frac{5}{4} e^{4x} + C\]

Finally, to isolate \(y\), divide the entire equation by \(e^{4x}\):

\[y = \frac{\frac{5}{4} e^{4x} + C}{e^{4x}}\]

\[y = \frac{5}{4} + C e^{-4x}\]

Since \(\frac{5}{4} = 1.25\), the general solution is:

\[y = 1.25 + C e^{-4x}\]

Applying Initial Condition to Find C

We are given the initial condition \(y(0) = 2.25\). This means that when \(x=0\), the value of \(y\) is \(2.25\). We substitute these values into our general solution to determine the specific value of the constant \(C\):

\[2.25 = 1.25 + C e^{-4(0)}\]

Since any number raised to the power of zero is \(1\) (\(e^0 = 1\)):

\[2.25 = 1.25 + C \cdot 1\]

\[2.25 = 1.25 + C\]

Now, solve for \(C\):

\[C = 2.25 - 1.25\]

\[C = 1\]

Final Solution for the Differential Equation

Substitute the value of \(C=1\) back into the general solution \(y = 1.25 + C e^{-4x}\):

\[y = 1.25 + 1 \cdot e^{-4x}\]

\[y = e^{-4x} + 1.25\]

This is the particular solution to the given differential equation that satisfies the initial condition \(y(0) = 2.25\).

Comparing this result with the provided options:

The derived solution \(y = e^{-4x} + 1.25\) matches option 2.