Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
The equation of the curve satisfying the differential equation $y_2(x^2 + 1) = 2xy_1$ passing through the point (0, 1) and having slope of tangent at $x = 0$ as 3(where $y_2$ and $y_1$ represents $2^{nd}$ and $1^{st}$ order derivative), then:
$y = f(x)$ is strictly increasing function
$y = f(x)$ is non-monotonic function
$y = f(x)$ has three distinct real roots
$y = f(x)$ has only one negative root
Step-by-Step Solution
Key Concept: Reduce the second-order equation to first order by recognizing it as a separable differential equation after division.
The equation $(x^2 + 1)\frac{d^2y}{dx^2} = 2x\frac{dy}{dx}$ is rewritten in separable form. Integrating $\int \frac{2y}{y(x^2+1)} dx = \int \frac{2x}{x^2+1} dx$ gives $\ln|y| = \ln|x^2 + 1| + \ell nc$. With initial condition $y_1 = c(x^2 + 1)$ and $c = 3$, we obtain $y = 3x^3 + 3x + 1$.
Correct Answer: 1,4