Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Find the general solution of the differential equation
$\text{x}\log\text{x}.\frac{\text{dy}}{\text{dx}}+\text{y}=\frac{2}{\text{x}}\cdot\log\text{x}$.
✓

Answer

The given differential equation can be written as
$\frac{\text{dy}}{\text{dx}}+\frac{1}{\text{x log x}}\text{y}=\frac{2}{\text{x}^{2}}$
$\text{I.F.}=\text{e}^{\int\frac{1}{\text{x log x}}\text{dx}}=\text{e}^{\log(\log x)}=\log\text{x}$
The solution is $y . \log x = \int\frac{2}{\text{x}^{2}}\cdot\log\text{x dx + c}$
OR, $y . \log x = 2$
$\Bigg[\log\text{x}\cdot\Big(\frac{-1}{\text{x}}\Big)+\int\frac{\text{dx}}{{\text{x}}^{2}}\Bigg]+\text{c}=2\Bigg[\frac{-\log\text{x}}{\text{x}}-\frac{\text{1}}{\text{x}}\Bigg]+\text{c}$
$\Rightarrow\text{y}\cdot\log\text{x}=-\frac{2}{\text{x}}[1+\log\text{x}]+\text{c}$

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

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$(x^2 + y^2)^2 = xy$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}$
→
Vitamins $A$ and $B$ are found in two different foods $F_1$ and $F_2$. One unit of food $F_1$ contains 2 units of vitamin $A$ and 3 units of vitamin $B$. One unit of food $F_2$ contains 4 units of vitamin $A$ and 2 units of vitamin $B$. One unit of food $F_1$ and $F_2$ cost Rs 50 and 25 respectively. The minimum daily requirements for a person of vitamin $A$ and $B$ is 40 and 50 units respectively. Assuming that anything in excess of daily minimum requirement of vitamin $A$ and $B$ is not harmful, find out the optimum mixture of food $\mathrm{F}_1$ and $\mathrm{F}_2$ at the minimum cost which meets the daily minimum requirement of vitamin A and B . Formulate this as a LPP.
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{(\text{x}-2)^2(\text{x}+3)}\ \text{dx}$
→
Evaluate the following integrals:
$\int\limits_{0}^{{\frac{\pi}{4}}}(\tan\text{x}+\cot\text{x})^{-2}\text{ dx}$
→
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+\text{y}=0,\text{y}(0)=1,\text{y}(0)=2$
Function $\text{y}=\text{xe}^\text{x}+\text{e}^{\text{x}}$
→
Find the probability of 4 turning up at least once in two tosses of a fair die.
→
The length x of a rectangle is decreasing at the rate of 5cm/ minute and the width y is increasing at the rate of 4cm/ minute. When x = 8cm and y = 6cm, find the rates of change of:
  1. The perimeter.
  2. The area of the rectangle.
→
Find the equation of the plane passing through the intersection of the planes.
$ \vec{r}.(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k})}=7,\text{ }\vec{r}.(2\hat{\text{i}}+5\hat{\text{j}}+3\hat{\text{k}})=9$and the point (2, 1, 3).
→
Solve the following differential equations:
$(\text{x}-1)\frac{\text{dy}}{\text{dx}}=2\text{x}^3\text{y}$
→