Question

The equation of the curves, satisfying the differential equation $\frac{d^{2}y}{d{x}^2}(x^2+1)=2x\frac{dy}{dx}$ passing through the point $(0,1)$ and having the slope of tangent at $x=0$ as $6$ is

• A. $y=2x^3+6x+1$
• B. $y=2x^3+4x+3$
• C. $y=3x^2+6x+1$
• D. None of these

Answer 1

The correct option is A $y=2x^3+6x+1$

Given $\ddot{y}-\frac{2x}{x^2+1}\dot{y}=0$

It is in the form of linear equation

The integration factor is $e^{\int -\frac{2x}{x^2+1}dx}=\frac{1}{x^2+1}$

Therefore the general solution is $\dot{y}\frac{1}{x^2+1}=\int 0dx=c$

$\Rightarrow \dot{y}=c(x^2+1)$

Given that at $x=0$, the value of $\dot{y}$ is $6$

$\Rightarrow c=6$

Therefore, the solution is $\frac{dy}{dx}=6(x^2+1)$

By integrating on both sides, we get $y=2x^3+6x+C$

It passes through point $(0,1)$, so we get $C=1$

Therefore, the general solution is $y=2x^3+6x+1$