Differential Equations
Differential Equation
nta_abhyas_2025
Grade 12
Question:
If $\int x\,e^x\,dx = f(x)$ and the solution of the differential equation $\frac{dy}{dx} = 1 + xy\,is\,y = ke^{f(x/2)} + Ce^x$, then the value of $k$ is equal to (where $C$ is the constant of integration)
Step-by-Step Solution
Key Concept: Use the integrating factor method for linear first-order differential equations
This is a linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$ where $P(x) = -x$ and $Q(x) = 1$. The integrating factor is $IF = e^{\int -x dx} = e^{-x^2/2}$. Multiplying both sides by the integrating factor and integrating: $y = e^{x^2/2}\int e^{-x^2/2}dx + Ce^{x^2/2}$. Comparing with the standard form, we find $k = 1$.
Correct Answer: 1