If the curve y = f(x) passes through the point (1, e) and satisfies the differential equat…

Question

If the curve $y = f(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \log_e x) dx$, $x > 0$, then $f(e)$ is equal to :

Accepted Answer

$\frac{dy}{dx}=y(2+\ln x)$

To solve this, first separate the variables by dividing both sides by $y$:

$\int \frac{dy}{y}=\int (2+\ln x)\, dx$

Now integrate both sides. The left side gives $\ln y$. For the right side, integrate term by term:

$\ln y=2x+x\ln x-x+C$

Combine like terms $2x-x=x$:

$\ln y=x+x\ln x+C$

Since it passes through $(1,e)$, put $x=1$ and $y=e$ in the equation. Then $\ln y=\ln e=1$ and $\ln 1=0$:

$1=1+0+C	o C=0$

So the required relation becomes:

$\ln y=x+x\ln x$

Taking antilog (exponential) on both sides:

$f(x)=y={e}^{x+x\ln x}$

Now substitute $x=e$. Also, $\ln e=1$:

$\Rightarrow f(e)={e}^{e+e}={e}^{2e}$

Solutions · 1