Download App

Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Show that $\text{y}=\text{e}^\text{x}(\text{A}\cos\text{x}+\text{B}\sin\text{x})$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$
✓

Answer

We have,
$\text{y}=\text{e}^\text{x}(\text{A}\cos\text{x}+\text{B}\sin\text{x})\ ...(1)$
$\frac{\text{dy}}{\text{dx}}\text{e}^\text{x}(\text{A}\cos\text{x}+\text{B}\sin\text{x})\ ...(1)$
$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x}[(\text{A}+\text{B})\cos\text{x}-(\text{A}-\text{B})\sin\text{x}]\ ...(2)$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\text{e}^\text{x}[(\text{A}+\text{B})\cos\text{x}-(\text{A}-\text{B})\sin\text{x}]+\text{e}^\text{x}[-(\text{A}-\text{B})\cos\text{x}]$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\text{e}^\text{x}[(\text{A}+\text{B})\cos\text{x}-(\text{A}-\text{B})\sin\text{x}]$
$2\text{y}+\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\frac{\text{dy}}{\text{dx}}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$
Hence, $\text{y}=\text{e}^\text{x}(\text{A}\cos\text{x}+\text{B}\sin\text{x})$ is the solution to the given differential equation.
$\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$

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 "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

Find the maximum and minimum of the following functions:

$f(x)=2 x^3-21 x^2+36 x-20$

→
Form the differential equation of the family of curves represented by the equation (a being the perimeter):$(2\text{x}-\text{a})^2-\text{y}^2=\text{a}^2$
→
Find the equation of the plane through the points $(3, 4, 1)$ and $(0, 1, 0)$ and parallel to the line $\frac{\text{x}+3}{2}=\frac{\text{y}-3}{7}=\frac{\text{z}-2}{5}.$
→
A bag contains $4$ white and $5$ black balls and another bag contains $3$ white and $4$ black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is taken out from the latter. Find the probability that the ball drawn is white.
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}$
→
What are the values of 'a' for which $f(x) = a^x$​​​​​​​ is increasing on $R?$
→
In a triangle OAB, $\angle\text{AOB}=90^\circ.$ If P and Q are points of trisection of AB, prove that $\text{OP}^2+\text{OQ}^2=\frac{5}{9}\text{AB}^2.$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$
→
Two sides of a parallelogram are $3 \hat{i}+4 \hat{j}-5 \hat{k}$ and $-2 \hat{j}+7 \hat{k}$. Find the unit vectors parallel to the diagonals.
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}\{\log\text{y}-\log\text{x}+1\}$
→