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:
$\text{cos}\bigg(\frac{\text{dy}}{\text{dx}}\bigg) = \text{a}(\text{a} \in \text{R}); \ \text{y}=1\text{when x = }0$
✓

Answer

Given: Differention equation $\text{cos}\bigg(\frac{\text{dy}}{\text{dx}}\bigg) = \text{a}(\text{a} \in \text{R}); \ \text{y}=1\text{when x = }0$
$\ \Rightarrow \ \frac{\text{dy}}{\text{dx}}=\text{cos}^{-1}\text{a}$
$\Rightarrow \ \text{dy}=\Big(\text{cos}^{-1}\text{a}\Big)\text{dx}$
Integrating both sides, $\int 1\ \text{dy}=\int\Big(\text{cos}^{-1}\text{a}\Big)\text{dx}$
$\Rightarrow \ \text{y}=\Big(\text{cos}^{-1}\text{a}\Big)\int1\ \text{dx}$
$\ \Rightarrow \ \text{y}=\Big(\text{cos}^{-1}\text{a}\Big)\ \text{x+c}\ .....\text{(i)}$
Now putting y = 1 when x = 0 in eq. (i), we get c = 1
Putting c = 1 in eq. (i), $y = (\cos ^{-1} a) x + 1$
$\Rightarrow\frac{\text{y-1}}{\text{x}}=\cos^{-1} \text{a}$
$\Rightarrow cos\frac{\text{y-1}}{\text{x}}=\text{a}$

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

Evaluate the following integrals:$\int\frac{1}{\sqrt{(1-\text{x}^2)\big\{9+\big(\sin^{-1}\text{x}\big)^2\big\}}}\text{ dx}.$
→
If the line has direction ratios 2, -1, -2 determine its direction Cosines.
→
Evaluate the following integrals:
$\int\limits^4_0|\text{x}-1|\text{dx}$
→
Evaluate the following definite integrals:
$\int_{1}^\limits{\text{e}}\frac{\text{e}^{\text{x}}}{\text{x}}(1+\text{x}\log\text{x})\text{dx}$
→
Examine the differentialiblilty of the function f defined by $\text{f(x)}=\begin{cases}2\text{x}+3 & \text{if}-3\leq\text{x}\leq-2\\\text{x}+1 & \text{if} -2\leq\text{x}\leq0\\\text{x}+2&\text{if}\ 0\leq\text{x}\leq1\end{cases}$
→
Determine P(E|F) in Exercises.
Two coins are tossed once, where
E : tail appears on one coin, F : one coin shows head.
→
Form the differential equation corresponding to $\text{y}^2=\text{a}(\text{b}-\text{x}^2)$ bt eliminating a and b.
→
If x < 1, then write the value of $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big)$ in terms of $\tan^{-1}\text{x.}$
→
Find the matrix A such that
$ \begin{bmatrix}4\\1\\3\end{bmatrix}\text{A}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}$
→
Let A and B be two independent events such that $P(A) = p_1$ and $P(B) = p_2​​​​​​​$​​​​​​​. Describe in words the events whose probabilities are:
$p_1 + p_2 = 2p_1p_2.$
→