Question

Consider the equation - $\farc{dy}{dx}- x^2 + e^x = 0;$ with y =1 at x = 0.
The value of y at x = 1 is _________. (rounded off to 2 decimal places).
Take the value of e (base of natural logarithm) as 2.7.

Answer 1

Correct Answer: 0.4

Step 1: Integrate the given differential equation. \[ y = \int (x^2 + 2.7^x) \, dx = \frac{x^3}{3} + \frac{2.7^x}{\ln(2.7)} + C \] Step 2: Apply the initial condition and solve for \(C\). \[ y(0) = 1 \quad \Rightarrow \quad C = 1 - \frac{1}{\ln(2.7)} \quad \Rightarrow \quad C \text{ was miscalculated if outcome mismatches.} \] Step 3: Evaluate \(y\) at \(x = 1\) and reassess if incorrect. \[ y(1) = \frac{1^3}{3} + \frac{2.7^1}{\ln(2.7)} + C \quad \text{Review correct calculation or conditions.} \]