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

Question

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

Function $\text{y}=\sin\text{x}+\cos\text{x}$

✓

Answer

We have,
$\text{y}=\sin\text{x}+\cos\text{x}\ ....(1)$
Differentiating both sides of (1) with respect to x, we get
$\frac{\text{dy}}{\text{dx}}=\cos\text{x}-\sin{\text{x}}...(2)$
Differentiating both sides of (2) with respect to x, we get
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=-\sin\text{x}-\cos\text{x}$
$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=-(\sin\text{x}+\cos\text{x})$
$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=-\text{y}$ [Using (1)]
$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{y}=0$
It is the given differential equation.
Therefore, $\text{y}=\sin\text{x}+\cos\text{x}$ satiesfies the given differential equation.
Also, when $\text{x}=0;\text{y}=\sin0+\cos0=1,\text{i.e.y}(0)=1.$
And, when $\text{x}=0;\text{y}'=\cos0-\sin0=1,\text{i.e.y'}(0)=1$
Hence $\text{y}=\sin\text{x}+\cos\text{x}$ 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 "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

Evaluate the following definite integrals:
$\int_{0}^\limits{\infty}\frac{1}{\text{a}^2+\text{b}^2\text{x}^2} \text{ dx}$

Evaluate the following integrals:

$\int\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)\text{dx}$

Show that the curves 2x = y² and 2xy = k cut at right angles, if k² = 8.

Prove that:
$\begin{vmatrix}\text{a}^2+1&\text{ab}&\text{ac}\\\text{ab}&\text{b}^2+1&\text{bc}\\\text{ca}&\text{cb}&\text{c}^2+1 \end{vmatrix}=1+\text{a}^2+\text{b}^2+\text{c}^2$

The odds against a certain event are 5 to 2 and the odds in favour of another event, independent to the former are 6 to 5. Find the probability that,

At least one of the events will occur,
None of the events will occur.

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b - ab, for all a, b ∈ S.
Prove that:

* is a binary operation on S.
* is commutative as well as associative.

Let $\text{A} = \text{R} - \left\{3\right\}, \text{B} = \text{R} - \left\{1\right\}. \text{Let f : A} \rightarrow \text{B}$ be defined by $\text{f(x)}\frac{\text{x - 2}}{\text{x - 3}}, \forall \text{ x} \in \text{A}.$ Show that f is bijective. Also, find

$\text{x, if f}^{-1} \text{(x) = 4}$
$\text{f}^{-1} (7)$

Evaluate the following integrals:
$\int^\limits{(\pi)^\frac{2}{3}}_{0}\sqrt{\text{x}}\cos^2\text{x}^{\frac{3}{2}}\text{ dx}$

Solve the following systems of homogeneous linear equations by matrix method:

2x - y + 2z = 0

5x + 3y - z = 0

x + 5y - 5z = 0