Solve the following differential equation dy dx -x - Vidyadip

Question

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

✓

Answer

We have
$\frac{\text{dy}}{\text{dx}}-\text{x}\log\text{x}$
$\Rightarrow\text{dy}=(\text{x}\log\text{x})$
Integrating boh sides we get,
$\int\text{dy}=\int(\text{x}\log\text{x})\text{dx}$
$\Rightarrow\text{y}=\int\text{x}\times\log\text{x dx}$
$\Rightarrow\text{y}=\log\text{x}\int\text{x dx}-\int\Big[\frac{\text{d}}{\text{dx}}(\log\text{x})\int\text{x dx}\Big]\text{dx}$
$\Rightarrow\text{y}=\log\text{x}\times\frac{\text{x}^2}{2}-\int\Big(\frac{1}{\text{x}}\times\frac{\text{x}^2}{2}\Big)\text{dx}$
$\Rightarrow\text{y}=\frac{1}{2}\text{x}^2\log\text{x}-\int\frac{\text{x}}{2}\text{ dx}$
$\Rightarrow\text{y}=\frac{1}{2}\text{x}^2\log\text{x}-\frac{\text{x}^2}{4}+\text{C}$
hence, $\text{y}=\frac{1}{2}\text{x}^2\log\text{x}-\frac{\text{x}^2}{4}+\text{C}$ is the solutin to 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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cot\text{x}=\text{x}^2\cot\text{x}+2\text{x}$

Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.

Solve the following differential equation:
x dy – y dx = $\sqrt{\text{x}^{2}+\text{y}^{2}}\text{ dx}$ .

In a hospital, there are 20 kidney dialysis machines and that the chance of any one of them to be out of service during a day is 0.02. Determine the probability that exactly 3 machines will be out of service on the same day.

Evaluate the following definite integrals:
$\int\limits_{1}^{2}\frac{\text{x}}{(\text{x}+1)(\text{x}+2)}\text{ dx}$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\text{a}\sin\text{x}+\text{b}\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$

Solve the matrix equations:
$\begin{bmatrix}\text{x}&1\end{bmatrix}\begin{bmatrix}1&0\\-2&-3\end{bmatrix}\begin{bmatrix}\text{x}\\5\end{bmatrix}=0$

Find a point on the parabola $y = (x − 3)^2,$ where the \tangent is parallel to the chord joining $(3, 0)$ and $(4, 1).$

There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?

If $\text{y}=\tan^{-1}\Big(\frac{\sqrt{1+\text{x}}-\sqrt{1-\text{x}}}{\sqrt{1+\text{x}}+\sqrt{1+\text{x}}}\Big),$ find $\frac{\text{dy}}{\text{dx}}.$