Question

Find the general solution of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)).

Hint:

Always ensure a linear differential equation is in its standard form \(\frac{dy}{dx} + P(x)y = Q(x)\) before identifying \(P(x)\) and \(Q(x)\). A common mistake is to forget to divide by the coefficient of the \(\frac{dy}{dx}\) term.

Answer 1

Step 1: Understanding the Concept:
The given equation is a first-order linear differential equation. We need to find its general solution, which includes an arbitrary constant of integration.
Step 2: Key Formula or Approach:
A first-order linear differential equation is of the form \(\frac{dy}{dx} + P(x)y = Q(x)\).
The solution is found using an integrating factor (I.F.), calculated as \(I.F. = e^{\int P(x)dx}\).
The general solution is then given by \(y \cdot (I.F.) = \int Q(x) \cdot (I.F.) dx + C\).
Step 3: Detailed Explanation:
The given differential equation is: \[ x\frac{dy}{dx} + 2y = x^2 \] First, we rewrite it in the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\) by dividing the entire equation by \(x\) (since \(x \neq 0\)).
\[ \frac{dy}{dx} + \frac{2}{x}y = x \] By comparing with the standard form, we have: \[ P(x) = \frac{2}{x} \] \[ Q(x) = x \] Next, we find the integrating factor (I.F.): \[ I.F. = e^{\int P(x)dx} = e^{\int \frac{2}{x}dx} \] \[ I.F. = e^{2\ln|x|} = e^{\ln(x^2)} = x^2 \] Now, we use the formula for the general solution: \[ y \cdot (I.F.) = \int Q(x) \cdot (I.F.) dx + C \] \[ y \cdot x^2 = \int x \cdot x^2 dx + C \] \[ yx^2 = \int x^3 dx + C \] Integrating the right-hand side: \[ yx^2 = \frac{x^4}{4} + C \] Finally, we solve for \(y\) by dividing by \(x^2\): \[ y = \frac{x^4}{4x^2} + \frac{C}{x^2} \] \[ y = \frac{x^2}{4} + \frac{C}{x^2} \] Step 4: Final Answer:
The general solution of the given differential equation is \(y = \frac{x^2}{4} + \frac{C}{x^2}\), where C is an arbitrary constant.