Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
For each of the differential equations given in find the general solution:
$(\text{x}+3\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y}(\text{y}>0)$
✓

Answer

$(\text{x}+3\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}+3\text{y}^2}$
$\Rightarrow\frac{\text{dx}}{\text{dy}}=\frac{\text{x}+3\text{y}^2}{\text{y}}=\frac{\text{x}}{\text{y}}+3\text{y}$
$\Rightarrow\frac{\text{dx}}{\text{dy}}-\frac{\text{x}}{\text{y}}=3\text{y}$
This is a linear differential equation of the form:
$\frac{​​\text{dx}}{\text{dy}}+\text{px}=\text{Q}\ (\text{where p}=-\frac{1}{\text{y}}\ \text{and}\ \text{Q}=3\text{y})$
$\text{Now, I.F}=\text{e}^{\int\text{pdy}}=\text{e}^{-\int\frac{\text{dy}}{\text{y}}}=\text{e}^{-\log\text{y}}=\text{e}^{\log\Big(\frac{1}{\text{y}}\Big)}=\frac{1}{\text{y}}.$
The general solution of the given differential equation is given by the relation,
$\text{x}\text{(I.F.)}=\int(\text{Q}\times\text{I.F.})\text{dy}+\text{C}$
$\Rightarrow\text{x}\times\frac{1}{​​\text{y}}=\int\Big(3\text{y}\times\frac{1}{​​\text{y}}\Big)​​\text{dy}+​​\text{C}$
$\Rightarrow\frac{\text{x}}{\text{y}}=3\text{y}+\text{C}$
$\Rightarrow\text{x}=3\text{y}^2+\text{Cy}$

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 "3 Marks Question" 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

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
→
Write the vector equation of the line passing through the point (1, -2, -3) and normal to the plane $\vec{\text{r}}.(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})=5.$
→
If $\tan^{-1}\text{x}+\tan^{-1}\text{y}=\frac{\pi}{4},$ then write the value of x + y + xy.
→
If D, E, F are the mid-points of side BC, CA and AB respectively of a triangle ABC, write the value of $\overrightarrow{\text{AD}}+\overrightarrow{\text{BE}}+\overrightarrow{\text{CF}}$.
→
If $\int\Big(\frac{\text{x}-1}{\text{x}^2}\Big)\text{e}^{\text{x}}\text{ dx}=\text{f(x)}\text{e}^{\text{x}}+\text{C},$ then write the value of f(x).
→
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $(2\vec{\text{a}}+\vec{\text{b}})\ \text{and}\ (\vec{\text{a}}-3\vec{\text{b}})$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
→
Give an example of a function which is continuos but not differentiable at at a point.
→
Find the point on the curve $y = x^2$​​​​​​​ where the slope of the tangent is equal to the x-coordinate of the point.
→
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
→
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
→