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

Using differentials, find the approximate values of the following:
$\sqrt{25.02}$
→
If $\text{y}=\text{e}^{\text{x}^{\text{e}^\text{x}}}+\text{x}^{\text{e}^{\text{e}^\text{x}}}+\text{e}^{\text{x}^{\text{x}^{\text{e}}}},$ prove that $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}^{\text{e}^\text{x}}}\times\text{x}^{\text{e}^{\text{x}}}\Big\{\frac{\text{e}^\text{x}}{\text{x}}+\text{e}^\text{x}\log\text{x}\Big\}+\text{e}^{\text{x}^{\text{e}^{\text{x}}}}\times\text{e}^{\text{e}^\text{x}}\Big\{\frac{1}{\text{x}}+\text{e}^\text{x}\times\log\text{x}\Big\}+\text{e}^{\text{x}^{\text{x}^\text{e}}}\text{x}^{\text{x}^{\text{e}}}\times\text{x}^{\text{e}-1}\Big\{\text{x}+\text{e}\log\text{x}\Big\}$
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}\{\log\text{y}-\log\text{x}+1\}$
→
If $(\cos\text{x})^{\text{y}}=(\tan\text{y})^{\text{x}},$ Prove that $\frac{\text{dy}}{\text{dx}}=\frac{\log\tan\text{y}-\text{y}\tan\text{x}}{\log\cos\text{x}-\text{x}\sec\text{y cosec y}}$
→
Using differentials, find the approximate values of the following:
$\sqrt{49.5}$
→
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
7x + 5y + 6z + 30 = 0 and 3x - y - 10z + 4 = 0
→
Solve the following initial value problems:
$\Big\{\text{x}\sin^2\Big(\frac{\text{y}}{\text{x}}\Big)-\text{y}\Big\}\text{dx + x dy}=0,\text{y}(1)=\frac{\pi}4$
→
Sketch the graph y = |x + 3|. Evaluate $\int\limits_{-6}^{0}|\text{x}-3|\text{dx} $ . What does this value of the integral represent on the graph.
→
Solve the following problem graphically:
Minimise or Maximise Z = 3x + 9y subject to the constraints:
$x + 3 y \leq 60$
$x + y \geq 10$
$x \leq y$
$x \geq 0 , y \geq 0$
→
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.

→