Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Show that $\text{y}=4\text{ax}$ is a solution of the differential equation $\text{y}=\text{x}\frac{\text{d}\text{y}}{\text{dx}}+\text{a}\frac{\text{dx}}{\text{dy}}.$
✓

Answer

We have,

$\text{y}=4\text{ax}\ ...(1)$

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

$2\text{y}\frac{\text{dy}}{\text{dx}}=4\text{a}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2\text{a}}{\text{y}}\ ...(2)$

now, differentiating both sided of (1) with respect to y, we get

$2\text{y}=4\text{a}\frac{\text{dy}}{\text{dx}}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{2\text{a}}\ ...(3)$

$\therefore\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\text{x}\Big(\frac{2\text{a}}{\text{y}}\Big)+\text{a}\Big(\frac{\text{y}}{2\text{a}}\Big)$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\frac{2\text{ax}}{\text{y}}+\frac{\text{y}}{2}$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\frac{\text{y}^2}{2\text{y}}+\frac{\text{y}}{2}$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{2}+\frac{\text{y}}{2}$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}=\text{y}$

$\Rightarrow\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\frac{\text{dy}}{\text{dx}}$

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 "5 Marks Questions" 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

Show that the matrix $\text{A}=\begin{bmatrix}5&3\\12&7\end{bmatrix}$ is root of the equation $A^2 - 12A - I = 0$.
→
Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{(1-\text{x})^2}{(1+\text{x}^2)^2}\text{dx}$
→
Find the foot of perpendicular from the point (2, 3, 4) to the line $\frac{4-\text{x}}{2}=\frac{\text{y}}{6}=\frac{1-\text{z}}{3}.$ Also, find the perpendicular distance from the given point to the line.
→
Assume that the probability that a bomb dropped from an aeroplane will strike a certain target is 0.2. If 6 bombs are dropped, find the probability that.
  1. exactly 2 will strike the target.
  2. at least 2 will strike the target.
→
A square piece of tin of side $18\ cm$ is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
→
If R and S are relations on a set A, then prove that:
  1. R and S are symmetric $\Rightarrow\ \text{R}\cap\text{S}$ and $\text{R}\cup\text{S}$ are symmetric
  2. R is reflexive and S is any relation $\Rightarrow\ \text{R}\cup\text{S}$ is reflexive.
→
Find a particular solution of the differential equation (x - y) (dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)
→
Evaluate the following integrals:
$\int(\text{e}^{\log\text{x}}+\sin\text{x})\cos\text{x dx}$
→
Evaluate the following integrals:
$\int\frac{\text{x}^2-1}{\text{x}^4+1}\ \text{dx}$
→
Find the angles of a triangle whose vertices are A (0, -1 ,-2), B(3, 1 ,4) and C(5 ,7 ,1).
→