Show that y=A 2x+B 2x is a solution of the differential equation … - Vidyadip

Question

Show that $\text{y}=\text{A}\cos2\text{x}+\text{B}\sin2\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\text{y}=0$

✓

Answer

We have,

$\text{y}=\text{A}\cos2\text{x}+\text{B}\sin2\text{x}\ ...(1)$

Differentiating both sides of equation (1) with respect to 3, we get

$\frac{\text{dy}}{\text{dx}}=-2\text{A}\sin2\text{x}-\text{B}\cos2\text{x}\ ...(2)$

Differentiating both sides of equation (2) with respect to 3, we get

$\frac{\text{d}^2\text{y}}{\text{dx}^2}=-4\text{A}\cos2\text{x}+4\text{B}\sin2\text{x}$

$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-4(\text{A}\cos2\text{x}+4\text{B}\sin2\text{x})$

$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=-4\text{y}$

$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\text{y}=0$

Hence, the given function is the solution to the given differential equation.

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

A diet is to contain at least $80$ units of vitamin A and $100$ units of minerals. Two foods $F_1$ and $F_2$ are available. Food $F_1$ costs $Rs 4$ per unit and $F_2$ costs $Rs 6$ per unit one unit of food $F_1$ contains $3$ units of vitamin A and 4 units of minerals. One unit of food $F_2$ contains 6 units of vitamin A and $3$ units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these foods and also meets the mineral nutritional requirements.

If $\text{A}=\begin{bmatrix}3&-5\\-4&2\end{bmatrix},$ then find $A^2 - 5A - 14I$. Hence, obtain $A^3$.

Evaluate the following intregals:
$\int\frac{4\text{x}^2+3}{(\text{x}^2+2)(\text{x}^2+3)(\text{x}^2+4)}\ \text{dx}$

Evaluate : $\int \frac{1}{(\sin \theta)(3+2 \cos \theta)} \cdot d \theta$

Integrate the following w. r. t. x:

$\frac{6 x^3+5 x^2-7}{3 x^2-2 x-1}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}+\sin\Big(\frac{\text{y}}{\text{x}}\Big)$

Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=5\text{x}^{\frac{3}{2}}-3\text{x}^{\frac{5}{2}},\text{x}>0$

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$

A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin C and 9 units of vitamin D. The vitamin contents of 1kg of food X and 1kg of food Y are given below:

	Vitamin A	Vitamin B	Vitamin C	Vitamin D
Food X	1	1	1	2
Food Y	2	1	3	1

One kg food X costs Rs. 5, whereas one kg of food Y costs Rs. 8.
Find the least cost of the mixture which will produce the desired diet.

verify that $\text{y}^2=4\text{a}(\text{x}+\text{a})$ is a solution of the differential equation $\Big\{1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}=2\text{x}\frac{\text{dy}}{\text{dx}}.$