Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS4 Marks
Question
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{y}=0,\text{y}(0)=2,\text{y}'(0)=0$

Function $\text{y}=\text{e}^\text{x}+\text{e}^{-\text{x}}$

✓

Answer

We have

$\text{y}=\text{e}^{\text{x}}+\text{e}^{\text{-x}} ...(1)$

Differentiating both sides of (1) with respect to x, we get

$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}-\text{e}^{\text{x}} ...(2)$

Differentiating both sides of (2) with respect to x, we get

$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\text{e}^{\text{x}}+\text{e}^{\text{-x}}$

$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=\text{y}$ [Using (1)]

$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-\text{y}=0$

It is the given differential equation.

Therefore, y = ex + e-x satisfies the given differential equation.

Also, when x = 0; = e0 + e0 = 1 + 1, i.e. y(0) = 2.

And, when x = 0; y1 = e0 - e0 = 1 - 1, i.e. y'(0) = 0

Hence, y = ex + e-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

If $\text{x}=\text{a}\cos\theta,\text{y}=\text{b}\sin\theta$ Show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{b}^4}{\text{a}^2\text{y}^3}$ 
→
Evaluate the following integrals:
$\int\frac{1}{(\text{x}^2+2\text{x}+10)^2}\text{ dx}$
→
Draw a rough sketch of the given curve y = 1 + |x + 1|, x = -3, x = 3, y = 0 and find the area of the region bounded by them, using integration.
→
A particle moves along the curve y = x2 + 2x. At what point(s) on the curve are the x and y coordinates of the particle changing at the same rate?
→
$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}}=\vec{\text{b }} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$
→
If  $\text{f}\text{(x)}=\begin{cases}\frac{2^\text{z+2}-16}{4^\text{x}-16}, &\text{if x} \neq 2\\\text{k}, & \text{x} = 2\end{cases}$ 
is continuous at x = 2, Find k.
→
Solve the following differential equation
$(\sin\text{x}+\cos\text{x})\text{dy}+(\cos\text{x}+\sin\text{x})\text{dx}=0$
→
Solve the differential equation:
$y + x \frac{dy}{dx} = x - y \frac{dy}{dx}$
→
If x = $ \sqrt { a ^ { \sin ^ { - 1 } t} }$, y = $ \sqrt { a ^ { \cos ^ { - 1 } t}}$, show that $ \frac { d y } { d x } = -\frac { y } { x }$.
→
Two biased dice are thrown together. For the first die $\text{P}(6)=\frac{1}{2},$ the other scores being equally likely while for the second die, $\text{P}(1)=\frac{2}{5}$ and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.
→