Solve the differential equation `x dy/dx + y = x^3y^6`

Answer

To solve the differential equation \( x \frac{dy}{dx} + y = x^3 y^6 \), we will follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ x \frac{dy}{dx} + y = x^3 y^6 \]
We can rearrange it to isolate the derivative term:
\[ x \frac{dy}{dx} = x^3 y^6 - y \]

### Step 2: Divide by \( y^6 \)
Next, we divide both sides by \( y^6 \) to simplify the equation:
\[ \frac{x}{y^6} \frac{dy}{dx} + \frac{y}{y^6} = x^3 \]
This simplifies to:
\[ \frac{x}{y^6} \frac{dy}{dx} + \frac{1}{y^5} = x^3 \]

### Step 3: Substitute \( v = y^{-5} \)
Let’s make the substitution \( v = y^{-5} \). Then, differentiating gives:
\[ \frac{dv}{dx} = -5y^{-6} \frac{dy}{dx} \]
Thus, we can express \( \frac{dy}{dx} \) in terms of \( v \):
\[ \frac{dy}{dx} = -\frac{1}{5} y^6 \frac{dv}{dx} \]

### Step 4: Substitute in the equation
Substituting \( v \) into the rearranged equation:
\[ \frac{x}{y^6} \left(-\frac{1}{5} y^6 \frac{dv}{dx}\right) + v = x^3 \]
This simplifies to:
\[ -\frac{x}{5} \frac{dv}{dx} + v = x^3 \]

### Step 5: Rearranging to standard linear form
Rearranging gives us:
\[ \frac{dv}{dx} + \frac{5v}{x} = -5x^2 \]
This is now in the standard form of a linear differential equation \( \frac{dv}{dx} + P(x)v = Q(x) \) where \( P(x) = \frac{5}{x} \) and \( Q(x) = -5x^2 \).

### Step 6: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{5}{x} \, dx} = e^{5 \ln |x|} = |x|^5 \]

### Step 7: Multiply through by the integrating factor
Multiplying the entire equation by \( |x|^5 \):
\[ |x|^5 \frac{dv}{dx} + 5|x|^4 v = -5|x|^3 \]

### Step 8: Integrate both sides
The left side can be rewritten as:
\[ \frac{d}{dx}(|x|^5 v) = -5|x|^3 \]
Integrating both sides:
\[ |x|^5 v = -\frac{5}{4}|x|^4 + C \]

### Step 9: Solve for \( v \)
Now, solving for \( v \):
\[ v = -\frac{5}{4|x|} + \frac{C}{|x|^5} \]

### Step 10: Substitute back for \( y \)
Recall \( v = y^{-5} \):
\[ y^{-5} = -\frac{5}{4|x|} + \frac{C}{|x|^5} \]
Taking the reciprocal gives:
\[ y^5 = \frac{1}{-\frac{5}{4|x|} + \frac{C}{|x|^5}} \]

### Final Result
Thus, the solution to the differential equation is:
\[ y = \left(-\frac{5}{4|x|} + \frac{C}{|x|^5}\right)^{-\frac{1}{5}} \]