Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
For each of the differential equation given in find the general solution:
$(1+\text{x}^2)\ \text{dy}+2\text{xy}\ \text{dx}=\cot\text{x}\ \text{dx}\ (\text{x}\neq0)$
✓

Answer

$(1+\text{x}^2)\ \text{dx}+2\text{xy}\ \text{dx}=\cot\text{xdx}$ $\Rightarrow\ \frac{\text{dy}}{\text{dx}}+\frac{2\text{xy}}{1+\text{x}^2}=\frac{\cot\text{xdx}}{1+\text{x}^2}$This equation is a linear dyfferential equation of the form:
$\frac{\text{dy}}{\text{dx}}+\text{py}=\text{Q}\ \big(\text{where p}=\frac{2\text{x}}{1+\text{x}^2}\ \text{and}\ \text{Q}=\frac{\cot\text{x}}{1+\text{x}^2}\big)$ $\text{Now, I.F}=\text{e}^{\int{\text{pdx}}}=\text{e}^{\int{\frac{2\text{x}}{1+\text{x}^2}\text{dx}}}=\text{e}^{\log(1+\text{x}^2)}=1+\text{x}^2.$ The general solution of the given differential equation is given by the relation, $\text{y}(\text{I.F})=\int(\text{Q}\times\text{I.F})\ \text{dx}+\text{C}$ $\Rightarrow\ ​​\text{y}(1+\text{x}^2)=\int\Big[\frac{\cot\text{x}}{1+\text{x}}\times(1+\text{x}^2)\Big]\text{dx}+\text{C}$ $\Rightarrow\ \text{y}(1+\text{x}^2)=\int\cot\text{x dx}+\text{C}$ $\Rightarrow\ \text{y}(1+\text{x}^2)=\log|\sin\text{x}|+\text{C}$

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

Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^3+8)}\ \text{dx}$
→
If $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+5\hat{\text{j}},\ 3\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}-6\hat{\text{j}}-\hat{\text{k}}$ respectively are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether $\vec{\text{AB}}$ and $\vec{\text{CD}}$ are collinear or not.
→
A company manufactures two articles A and B. There are two departments through which these articles are processed: (i) assembly and (ii) finishing departments. The maximum capacity of the first department is 60 hours a week and that of other department is 48 hours per week. The product of each unit of article A requires 4 hours in assembly and 2 hours in finishing and that of each unit of B requires 2 hours in assembly and 4 hours in finishing. If the profit is Rs. 6 for each unit of A and Rs 8 for each unit of B, find the number of units of A and B to be produced per week in order to have maximum profit.
→
Using integration, find the area of the region bounded by the line 2y = 5x + 7, x-axis and the lines x = 2 and x = 8.
→
Differentiate the following function with respect to x:$(\log\text{x})^{x} + \text{x}^{\log\text{x}}.$
→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\sqrt{\cos\text{x}-\cos^3\text{x}}(\sec^3\text{x}-1)\cos^2\text{x dx}$
→
Form the differential equation of the family of curves represented by the equation (a being the perimeter):$(2\text{x}-\text{a})^2-\text{y}^2=\text{a}^2$
→
Find the inverse of the following matrices:$\begin{bmatrix}2 & -1 & 1 \\ -1 & 2 & -1\\ 1 & -1 & 2 \end{bmatrix}$
→
If A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is:
  1. Symmetric and transitive only.
  2. Reflexive and transitive only.
  3. Symmetric only.
  4. Transitive only.
→
Find two positive integers $x$ and $y$ such that their sum is $35$ and the product $x^2 y^5$ is a maximum.
→