Differential equation d^2y dx^2 +y=0,y(0)=0,y'(0)=1Function y= - Vidyadip

Question

Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0,\text{y}(0)=0,\text{y}'(0)=1$Function $\text{y}=\sin\text{x}$

✓

Answer

We have,$\text{y}=\sin{\text{x}} ...(1)$
Differentiating both sides of (1) with respect to x, we get
$\frac{\text{dy}}{\text{dx}}=\cos\text{x} ...(2)$
Differentiating both sides of (2) with respect to x, we get
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=-\sin{\text{x}}$
$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=-\text{y}$ [Using (1)]
$\Rightarrow \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{y}=0$
it is the given differential equation.
Here, $\text{y}=\sin\text{x}$ satisfies the given differential equation; hence, it is a solution.
Also, when $\text{x}=0,\text{y}=\sin0=0,\text{i.e.},\text{y}(0)=0.$
And, when $\text{x}=0,\text{y}'=\cos 0=1,\text{i.e.,}\text{y}'(0)=1.$
Hence, $\text{y}=\sin\text{x}$ is the solution to the given initial value problem.

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that $\text{f}(\text{x})=\log\sin\text{x}$ is increasing on $\Big(0,\frac{\pi}{2}\Big)$ and decreasing on $\Big(\frac{\pi}{2},\pi\Big).$

The following relation are defined on the set of real numbers.
aRb if a - b > 0
Find whether these relations are reflexive, symmetric or transitive.

Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of exactly one ace.

If $\text{y}=\cos^{-1}(2\text{x})+2\cos^{-1}\sqrt{1-4\text{x}^2}, 0 <\text{x}<\frac{1}{2},$ find $\frac{\text{dy}}{\text{dx}}.$

A bag contains $6$ red and $8$ black balls and another bag contains $8$ red and $6$ black balls. A ball is drawn from the first bag and without noticing its colour is put in the second bag. A ball is drawn from the second bag. Find the probability that the ball drawn is red in colour.

Show that the differential equation $\left(x \cos \frac{y}{x}\right)( ydx + xdy )=\left(y \sin \frac{y}{x}\right)( xdy - ydx )$ s homogeneous and solve it.

If $\text{A}=\text{diag}\begin{pmatrix}2&-5&9\end{pmatrix},\text{ B}=\text{diag}\begin{pmatrix}1&1&-4\end{pmatrix}$ and $\text{C}=\text{diag}\begin{pmatrix}-6&3&4\end{pmatrix},$ find.
$2\text{A}+3\text{B}-5\text{C}$

Let S be the set of all rational numbers of the for $\frac{\text{m}}{\text{n}},$ where $\text{m}\in\text{Z}$ and n = 1, 2, 3. Prove that * on sdefined by a * b = ab is not a binary operation.

In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{x}^2-25}{\text{x}-5},&\text{x}\neq5\\\text{k},&\text{x}=5\end{cases}\text{at x} =5$

Write the following in the simplest form:
$\tan^{-1}\Big\{\frac{\sqrt{1+\text{x}^2}+1}{\text{x}}\Big\},\text{x}\neq0$