Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS4 Marks
Question
Solve the following differential equation:
$\text{y}^2\frac{\text{dx}}{\text{dy}}+\text{x}-\frac{1}{\text{y}}=0$
✓

Answer

Here, $\text{y}^2\frac{\text{dx}}{\text{dy}}+\text{x}-\frac{1}{\text{y}}=0$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}+\frac{\text{x}}{\text{y}^2}=\frac{1}{\text{y}^3}$
It is a linear differential equation. Comparing the equation with,
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
$\text{P}=\frac{1}{\text{y}^2},\text{Q}=\frac{1}{\text{y}^3}$
I.F. $=\text{e}^{\int\text{Pdy}}$
$=\text{e}^{\int\frac{1}{\text{y}^2}\text{dy}}$
$=\text{e}^{-\frac{1}{\text{y}}}$
Solution of the equation is given by,
$\text{x}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dy + C}$
$\text{x}\Big(\text{e}^{-\frac{1}{\text{y}}}\Big)=\int\frac{1}{\text{y}^3}\Big(\text{e}^{-\frac{1}{\text{y}}}\Big)\text{dy + C}$
Let $\text{e}^{-\frac{1}{\text{y}}}=\text{t}$
$\Rightarrow\frac{1}{\text{y}}=-\log\text{t}$
$\text{e}^{-\frac{1}{\text{y}}}\times\frac{1}{\text{y}^2}\text{dy = dt}$
$\text{x (t)}=\int\frac{1}{\text{y}}\text{dt + C}$
$=-\log+\text{dt + C}$
$=-\Big[\log\text{t}\times\int1\times\text{dt}-\int\Big(\frac{1}{\text{t}}\int1\times\text{dt}\Big)\text{dt}\Big]+\text{C}$
$=-\Big[\text{t}\log\text{t}-\int\frac{\text{t}}{\text{t}}\text{dt}\Big]+\text{C}$
$\text{x (t)}=-\text{t}\log\text{t + t + C}$
$\text{x (t)}=-\text{t}[\log\text{t}-1]+\text{C}$
$\text{x}=-\Big[-\frac{1}{\text{y}}-1\Big]\text{Ce}^{\frac{1}{\text{y}}}$
$\text{x}=\frac{1}{\text{y}}+1+\text{Ce}^{\frac{1}{\text{y}}}$
$\text{x}=\Big(\frac{1+\text{y}}{\text{y}}\Big)+\text{Ce}^{\frac{1}{\text{y}}}$

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 "4 Marks" 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

Solve the following determinant equations:
$\begin{vmatrix}1&1&\text{x}\\\text{p}+1&\text{p}+1&\text{p}+\text{x}\\3&\text{x}+1&\text{x}+2\end{vmatrix}=0$
→
Integrate the function in Exercise:
$\frac{5\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+10}}$
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{(\text{x}^2+4)(\text{x}^2+25)}\ \text{dx}$
→
Find the integrals of the function cos 2x cos 4x cos 6x
→
Solve the following differential equation:
$(\text{x}+\text{y})^2\frac{\text{dy}}{\text{dx}} = 1$
→
There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and ​F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds the she needs atleast 14kg of nitrogen and 14kg of phosphoric acid for her crop. If F1 costs Rs 6/kg and F2 costs Rs 5/kg, determine how much of each type of fertilizer should be used so that the nutrient requirements are met at minimum cost. What is the minimum cost?
→
Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=\frac{3}{10}\text{x}^4-\frac{4}{5}\text{x}^3-3\text{x}^2+\frac{36}{5}\text{x}+11$
→
Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.
→
Evaluate the following intregals:
$\int\frac{1}{(\sin\text{x}-2\cos\text{x})(2\sin\text{x}-\cos\text{x})}\ \text{dx}$
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{6\text{x}}{\pi}-4\sin^{2}\text{x}\text{ on }\Big[0,\frac{\pi}{6}\Big]$
→