Step 1: Understanding the Concept:
The given equation is a first-order linear differential equation. It can be solved using the integrating factor method or by recognizing the left side as the derivative of a product.
Step 2: Key Formula or Approach:
A first-order linear DE has the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\).
The integrating factor (I.F.) is given by \(I.F. = e^{\int P(x)dx}\).
The solution is given by \(y \cdot (I.F.) = \int Q(x) \cdot (I.F.) dx + C\).
Alternatively, notice that the left side, \(x\frac{dy}{dx} + y\), is the result of the product rule for differentiation applied to \(xy\), i.e., \(\frac{d}{dx}(xy)\).
Step 3: Detailed Explanation:
The given differential equation is:
\[ x\frac{dy}{dx} + y = x^4 \]
Method 1: Integrating Factor
First, write the equation in standard form by dividing by \(x\):
\[ \frac{dy}{dx} + \frac{1}{x}y = x^3 \]
Here, \(P(x) = \frac{1}{x}\) and \(Q(x) = x^3\).
Calculate the integrating factor:
\[ I.F. = e^{\int P(x)dx} = e^{\int \frac{1}{x}dx} = e^{\ln|x|} = |x| \]
Assuming \(x>0\) (since the initial condition is at \(x=1\)), we take \(I.F. = x\).
The solution is given by:
\[ y \cdot x = \int x^3 \cdot x \, dx + C \]
\[ xy = \int x^4 \, dx + C \]
\[ xy = \frac{x^5}{5} + C \]
\[ y = \frac{x^4}{5} + \frac{C}{x} \]
Now, use the initial condition \(y(1) = \frac{6}{5}\) to find \(C\).
\[ \frac{6}{5} = \frac{(1)^4}{5} + \frac{C}{1} \]
\[ \frac{6}{5} = \frac{1}{5} + C \]
\[ C = \frac{6}{5} - \frac{1}{5} = \frac{5}{5} = 1 \]
Substitute \(C=1\) back into the general solution:
\[ y = \frac{x^4}{5} + \frac{1}{x} \]
Method 2: Recognizing the Product Rule
The left side of the equation is \(x\frac{dy}{dx} + 1 \cdot y\), which is exactly \(\frac{d}{dx}(xy)\).
So, the equation can be written as:
\[ \frac{d}{dx}(xy) = x^4 \]
Integrate both sides with respect to \(x\):
\[ \int \frac{d}{dx}(xy) dx = \int x^4 dx \]
\[ xy = \frac{x^5}{5} + C \]
This leads to the same general solution as in Method 1, and applying the initial condition gives the same final result.
Step 4: Final Answer:
The solution to the initial value problem is \(y = \frac{x^4}{5} + \frac{1}{x}\), which matches option (B).