Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation
$\text{x}(\text{x}^{2} - 1)\frac{\text{dy}}{\text{dx}} = 1, \text{y}(2) = 0$
✓

Answer

We have
$\text{x}(\text{x}^{2} - 1)\frac{\text{dy}}{\text{dx}} = 1$
$\Rightarrow \frac{\text{dy}}{\text{dx}} = \frac{1}{\text{x}(\text{x}^{2} - 1)}$
$\Rightarrow\text{dy} = \bigg\{\frac{1}{\text{x}(\text{x}^{2} - 1)}\bigg\} \text{dx}$
Integrating both sides, we get
$\int \text{dy} = \int \bigg\{\frac{1}{\text{x}(\text{x}^{2} - 1)}\bigg\} \text{dx}$
$\Rightarrow \text{y} = \int\bigg\{\frac{1}{\text{x}(\text{x}^{2} - 1)}\bigg\} \text{dx}$
$\Rightarrow \text{y} = \int \frac{1}{\text{x}(\text{x} - 1)(\text{x} + 1)} \text{dx}$
$$Let $\frac{1}{\text{x}(\text{x} - 1)(\text{x} + 1)} = \frac{\text{A}}{\text{x}} + \frac{\text{B}}{\text{x}-1} + \frac{\text{C}}{\text{x}+ 1}$
$\Rightarrow 1 = \text{A} (\text{x}^{2} - 1)+\text{B} (\text{x}^{2} +\text{x) } + \text{C} (\text{x}^{2} - \text{x})$
$\Rightarrow 1 = (\text{A + B + C}) \text{x}^{2} + (\text{B} - \text{C}) \text{x} - \text{A}$
Equating the coefficients on both sides we get
A + B + C = 0 .....(1)
B - C = 0 .....(2)
A = -1 .....(3)
Solving (1), (2) and (3), we get
$\text{A} = -1$
$\text{B} = \frac{1}{2}$
$\text{C} = \frac{1}{2}$
$\therefore \text{y} = \frac{1}{2} \int \frac{1}{\text{x} - 1} \text{dx} - \int \frac{1}{\text{x}} \text{dx} + \frac{1}{2} \int \frac{1}{\text{x + 1}} \text{dx}$
$= \frac{1}{2} \log |\text{x } - 1| - \log|\text{x}| + \frac{1}{2}\log|\text{x + 1}| + \text{C}$
$= \frac{1}{2} \log |\text{x} - 1| + \frac{1}{2} \log | \text{x + 1} | - \log |\text{x}| + \text{C}$
It is given that y(2) = 0.
$\therefore 0 = \frac{1}{2} \log |2 - 1| + \frac{1}{2}\log|2+1|-\log|2|+\text{C}$
$\text{C}=\log|2|-\frac{1}{2}-\frac{1}{2}\log|3|$
Substituting the value of C, we get
$\text{y}=\frac{1}{2}\log|\text{x}-1|+\frac{1}{2}\log|\text{x}+1|-\log|\text{x}|+\log|2|-\frac{1}{2}\log|3|$
$\Rightarrow2\text{y}=\log|\text{x}-1|+\log|\text{x}+1|-2\log|\text{x}|+2\log|2|-\log|3|$
$\Rightarrow2\text{y}=\log|\text{x}-1+\log|\text{x}+1|-\log|\text{x}^3|+\log4-\log3$
$\Rightarrow2\text{y}=\log\frac{4(\text{x}-1)(\text{x}+1)}{3\text{x}^2}$
$\Rightarrow\text{y}=\frac{1}{2}\log\frac{4(\text{x}^2-1)}{3\text{x}^2}$
Hence, $\text{y}=\frac{1}{2}\log\frac{4(\text{x}^2-1)}{3\text{x}^2}$ is the solution to the given differerntial 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

find the area of the region bound by the curve $x = at^2, y = 2$ at between the ordinatrs corresponding $t = 1$ and $t = 2$.
→
Find the equation of the plane passing through the line of intersection of the planes $2x - 7y + 4z = 0, 3x - 5y + 4z + 11 = 0$ and the point $(-2, 1, 3).$
→
There are three coins. One is two headed coin, another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
→
Three urns contains 2 white and 3 black balls; 3 white and 2 black balls and 4 white and 1 black ball respectively. One ball is drawn from an urn chosen at random and it was found to be white. Find the probability that it was drawn from the first urn.
→
Solve the following differential equation:
$\frac{\text{y}}{\text{x}}\cos\Big(\frac{\text{y}}{\text{x}}\Big)\text{dx}-\Big\{\frac{\text{x}}{\text{y}}\sin\Big(\frac{\text{y}}{\text{x}}\Big)+\cos\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}=0$
→
If $y = \tan^{-1} \Bigg[\frac{\sqrt{1+\text{x}^{2}}-\sqrt{1-\text{x}^{2}}}{\sqrt{1+\text{x}^{2}}+\sqrt{1-\text{x}^{2}}}\Bigg],\text{ find }\frac{\text{dy}}{\text{dx}}.$
→
The scalar product of the vector $\hat{\text{l}}+\hat{\text{j}}+\hat{\text{k}}$ with a unit a vector along the sum of vectors $2\hat{\text{l}}+4\hat{\text{j}}-5\hat{\text{k}}\ \text{and}\ \lambda\hat{\text{l}}+2\hat{\text{j}}+3\hat{\text{k}}$ is equal to one. Find the value of $\lambda.$
→
Verify mean value theorem for the function:
$\text{f(x)}=\sqrt{25-\text{x}^2}\text{ in }[1,5].$
→
Evaluate the following integrals:
$\int\frac{\text{x}^2}{1-\text{x}^4}\ \text{dx}$
→
Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red find the probability that two red balls were transferred from A to B.
→