Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
For each of the differential equations in find a particular solution satisfying the given condition:
$\frac{\text{dy}}{\text{dx}}=\text{y}$ tan x; y = 1 when x = 0
✓

Answer

The given differential equation is
$\frac{\text{dy}}{\text{dx}}=\text{y tan x}$
Separating the variables, we get,

$\frac{1}{\text{y}}\text{dy}= \text{tanx}\ \text{dx}$
$\text{Integrating,}\ \int\frac{1}{\text{y}}\text{dy}=\int\text{tanx dx}$
$\therefore\ \log|\text{y}|=-\log|\cos{\text{x}}|+\text{c'}\ \ ... (1)$
$\therefore\ \log|\text{y}|+\log|\cos{\text{x}}|=\text{c'}$
$\therefore\ \log|\text{y}\cos{\text{x}}|=\text{c'}$
$\therefore|\text{y}\cos{\text{x}}|=\text{e}^{\text{c'}}$
$\therefore\ \text{y}\cos{\text{x}}=\pm\text{e}^{\text{c}}\ ​​\text{or}\ \text{y cosx}=\text{c}$
Now y = 1 when x = 0
$\therefore1\cos0=\text{c}\ \Rightarrow\ 1\times1=\text{c}\Rightarrow\text{c}=1$
$\therefore\text{form (1),}\ \text{y}\cos\text{x}=1,$ which is required solution.

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 "3 Marks Question" 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

Write the value of $\cos^{-1}\Big(\frac{1}{2}\Big)+2\sin^{-1}\Big(\frac{1}{2}\Big).$
→
If $y \sqrt{1-x^2}=\sin ^{-1} x$, then find $\frac{d y}{d x}$
→
Check the commutativity and associativity of the following binary operations:
'*' on Z defined by a * b = a + b - ab for all a, b ∈ Z.
→
If $\text{y}=\text{ax}^{\text{n+1}}+\text{bx}^{-\text{n}}$ and $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}=\lambda\text{y}$ then write the value of $\lambda$
→
Evaluate the following integrals:$\int\text{e}^{\text{x}}(\log\text{x}+\frac{1}{2})\text{dx}$
→
Find the direction cosines of the unit vector perpendicular to the plane $\vec{\text{r}}.\big(6\hat{\text{i}}-3\hat{\text{j}}-2\hat{\text{k}}\big)+1=0$ passing through the origin.
→
For the following differntial equations verify that the accompanying function is a solution:
Differential equation Function
$\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$ $\text{y}=\pm\sqrt{\text{a}^2-\text{x}^2}$
→
Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are $3$cm and $3.0005$cm, respectively.
→
The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?
→
Find the principal value of the following:
$\sec^{-1}\Big(2\tan\frac{3\pi}{4}\Big)$
→