Question

The general solution of the differential equation (x² + 2)dy +2xydx = e^x(x²+2)dx is

• A. \(\frac{x}{y}=e^x(x^2+x-4)+c\)
• B. 2xy = e^x(x²-2x+4)+c
• C. (x²+2)y=e^x(x²-2x+4)+c
• D. (x²+2)2y = e^x(x²+2x-4)+c

Answer 1

The Correct Option is C

Step 1: Rewrite the Equation in a Simpler Form

We begin by rewriting the equation in a more convenient form. Notice that the left-hand side resembles the derivative of a product. Specifically:

(x² + 2) dy + 2xy dx = d/dx (x² + 2)y

Thus, the equation becomes:

d/dx (x² + 2)y = e^x (x² + 2) dx

Step 2: Integrate Both Sides

Now that we have the equation in terms of a derivative, we can integrate both sides. Let’s do the integration step by step.

• Left-hand side: The integral of the derivative is just (x² + 2)y.
• Right-hand side: We need to integrate e^x (x² + 2) with respect to x. Let’s break it down into two simpler integrals:

        ∫ e^x (x² + 2) dx = ∫ e^x x² dx + ∫ 2 e^x dx    

Step 3: Integrate Each Term on the Right-Hand Side

1. First, solve ∫ e^x x² dx using integration by parts:

• Let u = x² and dv = e^x dx.
• Then, du = 2x dx and v = e^x.
• Apply the integration by parts formula: ∫ u dv = uv - ∫ v du

2. Next, solve ∫ 2x e^x dx using integration by parts:

• Let u = 2x and dv = e^x dx.
• Then, du = 2 dx and v = e^x.
• Apply the integration by parts formula again:

Now, putting everything together:

        ∫ e^x x² dx = x² e^x - (2x e^x - 2 e^x) = x² e^x - 2x e^x + 2 e^x    

Step 4: Combine Everything

Now combine all the terms from the integration:

        ∫ e^x (x² + 2) dx = e^x (x² - 2x + 4) + C    

Step 5: Write the Final Solution

Now, we can write the general solution. Recall the equation we had earlier:

        (x² + 2)y = e^x (x² - 2x + 4) + C    

Conclusion

We have solved the given differential equation and arrived at the general solution. By following these steps, you can solve similar first-order linear differential equations and understand the techniques involved in integrating complex expressions.