Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}\cos(\text{x}-\text{y})=1$
✓

Answer

$\frac{\text{dy}}{\text{dx}}\times\cos(\text{x}-\text{y})=1$
Let $\text{x}-\text{y}=\text{v}$
$1-\frac{\text{dy}}{\text{dx}}=\frac{\text{dv}}{\text{dx}}$
$\frac{\text{dy}}{\text{dx}}=1-\frac{\text{dv}}{\text{dx}}$
So,
$\Big(1-\frac{\text{dv}}{\text{dx}}\Big)\cos\text{v}=1$
$1-\frac{\text{dv}}{\text{dx}}=\sec\text{v}$
$1-\sec\text{v}=\frac{\text{dv}}{\text{dx}}$
$\text{dx}=\frac{\text{dv}}{1-\sec\text{v}}$
$\text{dx}=\frac{\cos\text{v}}{1-\cos\text{v}}\text{dv}$
$\int\text{dx}=\int\frac{\cos^{2}\frac{\text{v}}{2}-\sin^{2}\frac{\text{v}}{2}}{2\sin^{2}\frac{\text{v}}{2}}\text{dv}$
$\int\text{dx}=\int\frac{1}{2}\cot\big(\frac{\text{v}}{2}\big)\text{dv}-\frac{1}{2}\text{dv}$
$2\int\text{dx}=\int\cot^{2}\big(\frac{\text{v}}{2}\big)-\int\text{dv}$
$2\int\text{dx}=\int\Big(\text{cosec}^{2}\frac{\text{v}}{2}-1\Big)\text{dv}-\int\text{dv}$
$2\text{x}=-2\cot\big(\frac{\text{v}}{2}\big)\text{dv}-\text{v}-\text{v}+\text{C}_{1}$
$2(\text{x}+\text{v})=-2\cot\frac{\text{v}}{2}+\text{C}_{1}$
$\text{x}+\text{x}-\text{y}=-\cot\Big(\frac{\text{x}-\text{y}}{2}\Big)+\text{C}$
$\text{C}+\text{y}=\cot\Big(\frac{\text{x}-\text{y}}{2}\Big)$

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

If $A = \left[ {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ { - 1}&2&{ - 1} \\ 1&{ - 1}&2 \end{array}} \right]$, verify that $A^3 - 6A^2 + 9A - 4I = 0$ and hence find $A^{-1}$
→
Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}-\frac{\text{y}}{\text{x}}+\text{cosec}\frac{\text{y}}{\text{x}}=0,\text{y}(1)=0$
→
Find the matrix $X$ satisfying the equation:
$\begin{bmatrix}2 & 1 \\5 & 3 \end{bmatrix}\text{X}\begin{bmatrix}5 & 3 \\3 & 2 \end{bmatrix}=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}.$
→
Solve the differential equation $(\text{y}+3\text{x}^2)\frac{\text{dx}}{\text{dy}}=\text{x}$
→
Solve the following differential equations:
$\sqrt{1+\text{x}^2+\text{y}^2+\text{x}^2\text{y}^2}+\text{xy}\ \frac{\text{dy}}{\text{dx}}=0$
→
If $y^x = e^{y-x}$, Prove that $\frac{\text{dy}}{\text{dx}}=\frac{(1+\log\text{y})^2}{\log\text{y}}$
→
Show the solution zone of the following inequalities on a graph paper:
$\text{x}+\text{y}\leq50$
$3\text{x}+\text{y}\geq90$
$\text{x},\text{y}\geq0$
→
Evaluate the following integrals:
$\int\limits^2_{-1}\big(|\text{x}+1|+|\text{x}|+|\text{x}-1|\big)\text{dx}$
→
Integrate the function in Exercise:
$\frac{1}{\cos(\text{x}+\text{a)}\cos(\text{x}+\text{b)}}$
→
Let R be the relation over the set of all straight lines in a plane such that $\text{l}_1\text{Rl}_2\Leftrightarrow\text{l}_1\bot\text{l}_2.$ Then, R is:
  1. Symmetric.
  2. Reflexive.
  3. Transitive.
  4. An equivalence relation.
→