Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find $\text{y}=\text{Ae}^{-\text{kt}}\cos\text({pt}+\text{c})$prove that $\frac{\text{d}^2\text{y}}{\text{dt}^2}+2\text{k}\frac{\text{dy}}{\text{dt}}+\text{n}^2\text{y}=0,$Where $\text{n}^2=\text{p}^2+\text{k}^2.$
✓

Answer

We have,
$\text{y}=\text{Ae}^{-\text{kt}}\cos\text({pt}+\text{c})...(1)$
Differentiating y with respect to t, we get
$\frac{\text{dy}}{\text{dt}}=-\text{KAe}^{-\text{kt}}\cos(\text{pt}+\text{c})-\text{PAe}^{-\text{kt}}\sin(\text{pt}+\text{c})$
$=-\text{ky}-\text{PAe}^{-\text{kt}}\sin(\text{pt}+\text{c})\ [\text{from}(1)]$
$\Rightarrow\text{pAe}^{-\text{kt}}\sin(\text{pt}+\text{c})=-\text{ky}-\frac{\text{dy}}{\text{dt}}...(2)$
Differentiating $\frac{\text{dy}}{\text{dt}}$ with respect to t, we get
$\frac{\text{d}^2\text{y}}{\text{dt}}=-\text{k}\frac{\text{dy}}{\text{dt}}+\text{pkAe}^{-\text{kt}}\sin(\text{pt}+\text{c})-\text{p}^2\text{Ae}^{-\text{kt}}\cos(\text{pt}+\text{c})$
$=-\text{k}\frac{\text{dy}}{\text{dt}}+\text{k}\Big(-\text{ky}-\frac{\text{dy}}{\text{dt}}\Big)-\text{p}^2\text{y}\ [\text{from}(1)\ \text{and}\ (2)]$
$=-\text{k}\frac{\text{dy}}{\text{dt}}-\text{k}^2\text{y}-\text{k}\frac{\text{dy}}{\text{dt}}-\text{p}^2\text{y}$
$=-2\text{k}\frac{\text{dy}}{\text{dt}}-(\text{k}^2+\text{p}^2)\text{y}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dt}^2}+2\text{k}\frac{\text{dy}}{\text{dt}}+(\text{k}^2+\text{p}^2)\text{y}=0$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dt}^2}+2\text{k}\frac{\text{dy}}{\text{dt}} \text{n}^2\text{y}=0,$ where $\text{n}^2=\text{p}^2+\text{k}^2.$
Hence,
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dt}^2}+2\text{k}\frac{\text{dy}}{\text{dt}} \text{n}^2\text{y}=0,$ where $\text{n}^2=\text{p}^2+\text{k}^2.$

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 CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The probability distribution of a random variable X is given below:
$\text{X}$ $0$ $1$ $2$ $3$
$\text{P}(\text{X})$ $\text{k}$ $\frac{\text{k}}{2}$ $\frac{\text{k}}{4}$ $\frac{\text{k}}{8}$
  1. Determine the value of k.
  2. Determine $\text{P}(\text{X}\leq2)$ and $\text{P}(\text{X}\geq2)$
  3. Find $\text{P}(\text{X}\leq2)+\text{P}(\text{X}\geq2)$
→
A publisher sells a hard cover edition of a text book for Rs. 72.00 and paperback edition of the same ext for Rs. 40.00. Costs to the publisher are Rs. 56.00 and Rs. 28.00 per book respectively in addition to weekly costs of Rs. 9600.00. Both types require 5 minutes of printing time, although hardcover requires 10 minutes binding time and the paperback requires only 2 minutes. Both the printing and binding operations have 4,800 minutes available each week. How many of each type of book should be produced in order to maximize profit?
→
Find the general solution of $\left(1+x^{2}\right) d y+2 x y d x=\cot x d x(x \neq 0)$
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\sqrt{\frac{\text{y}^2}{\text{x}^2}-1}$
→
The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?
→
If $f(x) = x^2 - 2x$, find f(A), where $\text{A}=\begin{bmatrix}0&1&2\\4&5&0\\0&2&3\end{bmatrix}$
→
Using Lagrange's mean value theorem, prove that
$(\text{b}-\text{a})\sec^2\text{a}<\tan\text{b}-\tan\text{a}<(\text{b}-\text{a})\sec^2\text{b}$
where $0<\text{a}<\text{b}<\frac{\pi}{2}.$
→
Evaluate the following integrals as limit of sum:
$\int\limits^{2}_{0}\big(\text{x}^2+2\big)\text{dx}$
→
Sketch the region $\{(x, y): 9x^2+ 4y^2 = 36\}$ and find the area of the enclosed by it, using integration.
→
If $\text{y}=\sin(\sin\text{x})$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\tan\text{x}.\frac{\text{dy}}{\text{dx}}+\text{y}\cos^2\text{x}=0$
→