The particular solution of the differential equation `y'+3xy=x` which passes through (0,4) is

Options

A. `y=1-11e^(-3x^(2)//2)`
B. `3y=1+11e^(-3x^(2)//2)`
C. `3y=1-11e^(-3x^(2)//2)`
D. none of these

Answer

To solve the differential equation \( y' + 3xy = x \) that passes through the point (0, 4), we will follow these steps:

### Step 1: Identify the form of the differential equation
The given equation is a first-order linear differential equation of the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = 3x \) and \( Q(x) = x \).

**Hint:** Recognize the standard form of a linear differential equation.

### Step 2: Calculate the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int 3x \, dx} = e^{\frac{3x^2}{2}} \]

**Hint:** The integrating factor is derived from the function \( P(x) \) in the equation.

### Step 3: Multiply the entire differential equation by the integrating factor
Multiply the equation by \( \mu(x) \):
\[ e^{\frac{3x^2}{2}} \frac{dy}{dx} + 3x e^{\frac{3x^2}{2}} y = x e^{\frac{3x^2}{2}} \]

**Hint:** This step helps to simplify the left-hand side into a single derivative.

### Step 4: Rewrite the left-hand side as a derivative
The left-hand side can be rewritten as:
\[ \frac{d}{dx} \left( y e^{\frac{3x^2}{2}} \right) = x e^{\frac{3x^2}{2}} \]

**Hint:** Use the product rule of differentiation to combine terms.

### Step 5: Integrate both sides
Integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx} \left( y e^{\frac{3x^2}{2}} \right) dx = \int x e^{\frac{3x^2}{2}} \, dx \]
This gives:
\[ y e^{\frac{3x^2}{2}} = \int x e^{\frac{3x^2}{2}} \, dx + C \]

**Hint:** The right-hand side requires integration by substitution.

### Step 6: Solve the integral on the right-hand side
Let \( t = \frac{3x^2}{2} \), then \( dt = 3x \, dx \) or \( dx = \frac{dt}{3x} \). The integral becomes:
\[ \int x e^{\frac{3x^2}{2}} \, dx = \int e^{t} \frac{dt}{3} = \frac{1}{3} e^{t} + C = \frac{1}{3} e^{\frac{3x^2}{2}} + C \]

**Hint:** Use substitution to simplify the integral.

### Step 7: Substitute back into the equation
Substituting back, we have:
\[ y e^{\frac{3x^2}{2}} = \frac{1}{3} e^{\frac{3x^2}{2}} + C \]

### Step 8: Solve for \( y \)
Now, divide both sides by \( e^{\frac{3x^2}{2}} \):
\[ y = \frac{1}{3} + Ce^{-\frac{3x^2}{2}} \]

**Hint:** Isolate \( y \) to express it in terms of \( C \).

### Step 9: Use the initial condition to find \( C \)
We know that the solution passes through the point (0, 4):
\[ 4 = \frac{1}{3} + Ce^{0} \implies 4 = \frac{1}{3} + C \implies C = 4 - \frac{1}{3} = \frac{12}{3} - \frac{1}{3} = \frac{11}{3} \]

**Hint:** Substitute the initial condition to find the constant \( C \).

### Step 10: Write the particular solution
Substituting \( C \) back into the equation gives:
\[ y = \frac{1}{3} + \frac{11}{3} e^{-\frac{3x^2}{2}} \]

**Final Result:**
The particular solution of the differential equation that passes through (0, 4) is:
\[ y = \frac{1}{3} + \frac{11}{3} e^{-\frac{3x^2}{2}} \]

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