Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
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 "Solve the Following Question.(5 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{x}=\cos\text{t}+\log\tan\frac{\text{t}}{2},\text{y}=\sin\text{t},$ Then find the value of $\frac{\text{d}^2\text{y}}{\text{dt}^2}\ \text{and}\ \frac{\text{d}^2\text{y}}{\text{dx}^2}\ \text{at}\ \text{t}=\frac{\pi}{4}.$
→
A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is $2\ m$ and volume is $8m^3.$ If building of tank cost $70$ per square metre for the base and Rs $45$ per square matre for sides, what is the cost of least expensive tank?
→
Show that the following system of linear equations is consistent and also find solution:
$6x + 4y = 2$
$9x + 6y =3$
→
Consider $f : R \rightarrow R$ given by $f(x) = 4x + 3$. Show that f is invertible. Find the inverse of f.
→
Differentiate $\tan^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big)$ with respect to $\sqrt{1-\text{x}^2},$ if -1 < x < 1.
→
Solve the equation
$\cos^{-1}\frac{\text{a}}{\text{x}}-\cos^{-1}\frac{\text{b}}{\text{x}}=\cos^{-1}\frac{1}{\text{b}}-\cos^{-1}\frac{1}{\text{b}}-\cos^{-1}\frac{1}{\text{a}}$
→
Evaluate the following integrals as limit of sum:$\int\limits^1_{-1}(\text{x}+3)\text{dx}$
→
In a large bulk of items, $5$ percent of the items are defective. What is the probability that a sample of $10$ items will include not more than one defective item?
→
A small firm manufacturers items A and B. The total number of items A and B that it can manufacture in a day is at the most $24$. Item A takes one hour to make while item B takes only half an hour. The maximum time available per day is $16$ hours. If the profit on one unit of item A be $Rs. 300$ and one unit of item B be $Rs. 160$, how many of each type of item be produced to maximize the profit? Solve the problem graphically.
→
Show that the following system of linear equations is consistent and also find solutions:
$5x +3y + 7z = 4$
$3x + 26y + 2z = 9$
$7x + 2y + 10z = 5$
→