Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations4 Marks
Question
Solve the following differential equation
$\sqrt{1-\text{x}^4}\text{dy}=\text{x dx}$
✓

Answer

We have,
$\sqrt{1-\text{x}^4}\text{dy}=\text{x dx}$
$\Rightarrow\text{dy}=\frac{\text{x}}{\sqrt{1-\text{x}^4}}\ \text{dx}$
Integrating both sides, we get
$\int\text{dy}=\int\frac{\text{x}}{\sqrt{1-\text{x}^4}}\text{ dx}$
$\Rightarrow\text{y}=\int\frac{\text{x}}{\sqrt{1-\text{x}^4}}\ \text{dx}$
Putting x2 = t
$\Rightarrow2\text{x dx}=\text{dt}$
$\therefore\text{y}=\frac{1}{2}\int\frac{\text{dt}}{\sqrt{1-\text{t}^2}}$
$=\frac{\sin^{-1}}{2}+\text{C}$
$=\frac{1}{2}\sin^{-1}(\text{x}^2)+\text{C}$
hence, $\text{y}=\frac{1}{2}\sin^{-1}(\text{x}^2)+\text{C}$ is the solution to the given differential equation.

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 "4 Marks" 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

Solve the following systems of linear equations by cramer's rule:
x + y + z + w = 2,
x - 2y + 2z + 2w = -6,
2x + y - 2z + 2w = -5,
3x - y + 3z - 3w = -3
→
Evaluate the following intregals:
$\int\frac{2\text{x}+5}{\sqrt{\text{x}^2+2\text{x}+5}}\text{dx}$
→
Solve the following system of equations by matrix method:
x + y + z = 6
x + 2z = 7
3x + y + z = 12
→
The probability of a shooter hitting a target is $\frac{3}{4}.$ How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?
→
Find the intervals in which the following functions are increasing or decreasing.
f(x) = 8 + 36x + 3x2 -2x3
→
Evaluate the following integrals:
$\int\limits^{\pi}_2\log(1-\cos\text{x})\text{dx}$
→
For the matrix $\text{A}=\begin{bmatrix}1 & -1 & 1 \\2 & 3 & 0 \\ 18 & 2 & 10 \end{bmatrix},$ show that A (adjoint A) = 0.
→
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
f(x) = x(x + 4)2 on [0,4]
→
Show that the points whose position vectors are

$\vec{\text{a}}=4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}, \vec{\text{b}}=2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-\hat{\text{j}}$ from a right triangle.

→
Let X denot the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\text{P}(\text{X = x})=\begin{cases}\text{kx},&\text{if}\text{ x}=0\text{ or }1\\2\text{kx},&\text{if x = 2}\\\text{k}(5-\text{x}),&\text{if x = 3 or 4}\\0,&\text{if x > 4}\end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get addmission in
  1. Exactly one college.
  2. At most two colleges.
  3. At least two colleges.
→