Download App

Higher Order Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives5 Marks
Question
Find A and B so that $\text{y}=\text{A}\sin3\text{x}+\text{B}\cos3\text{x}$ satisfy the equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\frac{\text{dy}}{\text{dx}}+3\text{y}=10\cos3\text{x}.$
✓

Answer

$\text{y}=\text{Ae}^{-kt}\cos(\text{pt}+\text{c})$ Differentiating w.r.t.x, 
 $\Rightarrow\frac{\text{dy}}{\text{dt}}=\text{A}\Big\{\text{e}^{-\text{kt}}(-\sin(\text{pt}+\text{c})\times\text{p})+(\cos(\text{pt}+{c}))(-\text{re}^{-\text{kt}})\Big\}$
$\Rightarrow-\text{Ape}^{-\text{kt}}\sin(\text{pt}+\text{c})-\text{KAe}^{-\text{kt}}\cos(\text{pt}+\text{c})$
$\Rightarrow\frac{\text{dy}}{\text{dt}}=-\text{Ape}^{-\text{kt}}\sin(\text{pt}+\text{c})-\text{ky}$
Differentiating w.r.t.x,
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dt}^2}=-\text{Ap}\Big\{\text{e}^{-\text{kt}}(\cos(\text{pt}+\text{c})\times\text{p})+(\sin(\text{pt}+\text{c}))(\text{e}^{-\text{kt}}\times-\text{R})-\text{ky}^1\Big\}$$=-\text{p}^2\text{y}+\text{Apke}^{-\text{kt}}\sin(\text{pt}+\text{c})-\text{ky}^1$
Adding & substracting $ky_1$ on
RHS $\Rightarrow\frac{\text{d}^2\text{y}}{\text{dt}^2}=+\text{Apke}^{-\text{kt}}\sin(\text{pt}+\text{c})-\text{p}^2\text{y}-2\text{ky}^1+\text{ky}^1$
$\frac{\text{d}^2\text{y}}{\text{dt}^2}=\text{Apke}^{-\text{kt}}\sin(\text{pt}+\text{c})-\text{p}^2\text{y}-2\text{ky}^1-\text{k}\text{ape}^{-\text{kt}}\sin(\text{pt}+\text{c})-\text{k}^2\text{y}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dt}^2}=-(\text{p}+\text{k}^2)\text{y}-2\text{k}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dt}^2}+2\text{k}\frac{\text{dy}}{\text{dt}}+\text{n}^2\text{y}=0$

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 Higher Order Derivatives→Higher Order Derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\text{If x = }\sqrt{\text{a}^{\sin^{-1}t},}\text{ y}=\sqrt{\text{a}^{\text{cos}^{-1}}},\text{ show that }\frac{\text{dy}}{\text{dy}}=-\frac{\text{y}}{\text{x}}.$
→
A manufacturer makes two products A and B. Product A sells at Rs. 200 each and takes 1/2 hour to make. Product B sells at Rs. 300 each and takes 1 hour to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hours of production and weekly turnover must not be less than Rs 10000. If the profit on each of product A is Rs. 20 and on product B is Rs. 30, then how many of each should be produced so that the profit is maximum. Also, find the maximum profit.
→
If the product of the distances of the point (1, 1, 1) from the origin and the plane $x - y + z + λ = 0$ be 5, find the value of λ.
→
If $\text{A}=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then prove that $A^2 - A + 2I = O$.
→
Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\frac{\text{dy}}{\text{dx}}-\frac{2\text{xy}}{1+\text{x}^2}=\text{x}^2+2$
→
If a white ball is selected, what is the probability that it came from:
  1. Bag 2
  2. Bag 3
→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\frac{\tan\text{x}}{1+\text{m}^2\tan^2\text{x}}\text{ dx}$
→
By computing the shortest distance determine whether the following pairs of lines intersect or not:
$\vec{\text{r}}=\big(\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=\big(2\hat{\text{i}}-\hat{\text{j}}\big)+\mu\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$
→
Show that the differential equation $2ye^{x/y} dx + (y - 2xe^{x/y} ) dy = 0$ is homogeneous. Find the particular solution of this differential equation, given that $x = 0$ when $y = 1.$
→
Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{1+\text{x}}{(2+\text{x})^2}\text{dx}$
→