Solve the following differential equation dy dx = - Vidyadip

Question

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\log\text{x}$

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}=\log\text{x}$
$\Rightarrow\text{dy}=(\log\text{x})\text{dx}$
Integrating both sides, we get
$\int\text{dy}=\int(\log\text{x})\text{dx}$
$\Rightarrow\text{dy}=\int1\times\log\text{x}\text{ dx}$
$\Rightarrow\text{dy}=\log\text{x}\int\int1\text{dx}-\int\Big[\frac{\text{d}}{\text{dx}}(\log\text{x})\int1\text{dx}\Big]\text{dx}$
$\Rightarrow\text{y}=\text{x}\log\text{x }-\int\frac{\text{x}}{\text{x}}\text{dx}$
$\Rightarrow\text{y}=\text{x}\log\text{x}-\int1\text{dx}$
$\Rightarrow\text{y}=\text{x}\log\text{x}-\text{x}$
$\Rightarrow\text{y}=\text{x}(\log\text{x}-1)+\text{C}$
So, $\Rightarrow\text{y}=\text{x}(\log\text{x}-1)+\text{C}$ is defined for all $\text{x}\in\text{R}$ except x = 0
Hence, $\Rightarrow\text{y}=\text{x}(\log\text{x}-1)+\text{C}$ where $\text{x}\in\text{R}-\{0\}$ is the solution o the given differential equation.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the follwing intregals:
$\int\frac{1}{\text{x}^4-1}\text{ dx}$

Evalute the following integrals:
$\int\tan2\text{x}\tan3\text{x}\tan5\text{x dx}$

Verify Rolle's theorem for the following function on the indicated intervals $f(x) = x(x^- 4)^2$ on the interval $[0, 4]$

In order to supplement daily diet, a person wishes to take $X$ and $Y$ tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in $X$ and $Y$ are given as below:

Tablets	Iron	Calcium	Vitamin
$X$	$6$	$3$	$2$
$Y$	$2$	$3$	$4$

The person needs to supplement at least $18$ milligrams of iron, $21$ milligrams of calcium and $16$ milligrams of vitamins. The price of each tablet of $X$ and $Y$ is $₹ 2$ and $₹1$ respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

$\int\bigg[\log \log \text{x} + \frac{1}{(\log \text{x)}^{2}}\bigg]\text{dx}$

Using vectors find the area of the triangle with vertices, A(2, 3, 5), B(3, 5, 8) and C(2, 7, 8).

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\text{x}+\frac{1}{\text{x}}\text{ on }[1,3]$

If $e^x + x^y = e^{x+y},$ prove that $\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}+4\hat{\text{j}}+5\hat{\text{k}}\big)+\mu\big(3\hat{\text{i}}+4\hat{\text{j}}+5\hat{\text{k}}\big)$

Differentiate the following functions with respect to x:
$\sin^{-1}\big(1-2\text{x}^2\big),0<\text{x}<1$