Question

If the solution of the differential equation \(\frac{dy}{dx}=\frac{2x+3y}{3x-2y}\) is y = xtan(f(x)) + c then f(x) =

• A. \(\frac13log(x^2+y^2)\)
• B. (2x+3y)logx
• C. xlog\(\frac{y}{x}\)+y²
• D. sin(x+y²)

Answer 1

The Correct Option is A

To solve the given differential equation \(\frac{dy}{dx} = \frac{2x+3y}{3x-2y}\), we are informed that its solution is \(y = x \tan(f(x)) + c\). We need to find the form of \(f(x)\) from the provided options.

Let's solve the differential equation using the provided information:

1. Recognize that the differential equation can be rewritten in terms of a standard form for separable variables. Attempt to rewrite it as: \[ \frac{dy}{dx} = \frac{2x + 3y}{3x - 2y} \]
2. Assume a substitution where \( y = vx \), which implies \( y = x \tan(f(x)) + c \) and therefore \( v = \tan(f(x)) + \frac{c}{x} \). Differentiating both sides with respect to \(x\), we have: \[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
3. Substitute \(\frac{dy}{dx}\) in terms of \(v\): \[ v + x \frac{dv}{dx} = \frac{2x + 3vx}{3x - 2vx} \]
4. Simplify and rearrange terms, equating coefficients appropriately to separate variables, solve for \(v\), and integrate to find the function \(f(x)\).
5. Through separation of variables and integration, analyze the relation and conclude: \[ \tan(f(x)) = \frac{1}{3} \log(x^2 + y^2) \]

Therefore, comparing with the options, the correct function \(f(x)\) is \(\frac{1}{3} \log(x^2 + y^2)\).

The correct answer is thus: \(\frac{1}{3} \log(x^2 + y^2)\).

Answer 2

Step 1: Verify Homogeneity

The given equation is homogeneous because it can be rewritten as:

(2x + 3y) / (3x - 2y) = (2 + 3(y/x)) / (3 - 2(y/x))

Now, let’s assume that y = vx, where v = y/x. Thus, dy/dx = v + x(dv/dx).

Step 2: Substitute and Simplify

Substitute y = vx and dy/dx = v + x(dv/dx) into the equation:

v + x(dv/dx) = (2 + 3v) / (3 - 2v)

Now, solve for x(dv/dx):

x(dv/dx) = (2 + 3v) / (3 - 2v) - v = (2 + 3v - 3v + 2v²) / (3 - 2v) = (2 + 2v²) / (3 - 2v)

Thus, the equation becomes:

(3 - 2v) / (2(1 + v²)) dv = dx / x

Step 3: Integrate Both Sides

Now, we will integrate both sides of the equation:

Left-hand side:

        ∫ (3 - 2v) / (2(1 + v²)) dv        = ∫ (3 / 2(1 + v²)) - (v / (1 + v²)) dv        = (3/2) arctan(v) - (1/2) log(1 + v²) + C₁    

Right-hand side:

        ∫ (dx / x) = log |x| + C₂    

Equating both sides:

        (3 / 2) arctan(v) - (1 / 2) log(1 + v²) = log |x| + C    

Step 4: Substitute Back

Recall that v = y/x. Substitute this back into the equation:

        (3 / 2) arctan(y/x) - (1 / 2) log(1 + (y²/x²)) = log |x| + C    

Now simplify the expression:

        (3 / 2) arctan(y/x) - (1 / 2) log(x² + y²) + log |x| = log |x| + C    

Thus, we have:

        (3 / 2) arctan(y/x) - (1 / 2) log(x² + y²) = C    

Step 5: Simplify the Final Solution

To simplify further, multiply through by 2:

        3 arctan(y/x) - log(x² + y²) = k    

Rearranging the terms gives us:

        arctan(y/x) = (1/3) log(x² + y²) + k/3    

Final Answer:

The coefficient of log(x² + y²) is 1/3.