Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations4 Marks
Question
Solve the following initial value problems:
$(\text{x}^2+\text{y}^2)\text{dx}=2\text{xy dy, y}(1)=0$
✓

Answer

$(\text{x}^2+\text{y}^2)\text{dx}=2\text{xy dy, y}(1)=0$

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

It is a homogeneous equation.

Put y = vx
and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}},$

So,

$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x}^2+\text{v}^2\text{x}^2}{2\text{xvx}}$

$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{1+\text{v}^2}{2\text{v}}-\text{v}$

$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{1+\text{v}^2-2\text{v}^2}{2\text{v}}$

$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{1-\text{v}^2}{2\text{v}}$

$\int\frac{2\text{v}}{1-\text{v}^2}=\int\frac{\text{dx}}{\text{x}}$

$\log|1-\text{v}^2|=-\log|\text{x}|+\log|\text{C}|$

$\log|1-\text{v}^2|=\log\Big|\frac{\text{C}}{\text{x}}\Big|$

$\Big|\frac{\text{x}^2-\text{y}^2}{\text{x}^2}\Big|=\Big|\frac{\text{C}}{\text{x}}\Big|$

$|\text{x}^2-\text{y}^2|=|\text{Cx}|\ \dots(\text{i})$

Put y = 0, x = 1

1 - 0 = C

C = 1

Put the value of C in equation (i),

$|\text{x}^2-\text{y}^2|=|\text{x}|$

$(\text{x}^2-\text{y}^2)^2=\text{x}^2$

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

$\text{If y} = \tan^{-1} \bigg(\frac{\sqrt{1 + x^{2}}+{\sqrt{1 - x^{2}}}}{\sqrt{1 + x^{2}} - {\sqrt{1 - x^{2}}}}\bigg), x^{2}\leq 1, \text{then find} \frac{dy}{dx}.$

 

→
Find the distance of the point (-1, -5, -10) from the point of intersection of the line $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}+\lambda\Big(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}}\Big)$ and the plane $\vec{\text{r}}.\Big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\Big)=5.$
→
Evaluate the following integrals:
$\int(\text{x}-3)\sqrt{\text{x}^2+3\text{x}-18}\text{dx}$
→
Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g: A → B be functions defined by $f(\text{x})=\text{x}^2-\text{x},\ \text{x}\in\text{A}\ \text{and }\text{g(x)}=2\Big|\text{x}-\frac{1}{2}\Big|,\ \text{x}\in\text{A}.$ Are f and g equal? Justify your answer. (Hint: One may note that two functions f: A → B and g: A → B such that $f(\text{a}) = \text{g(a)}\ \forall \text{a} \in \text{A},$ are called equal functions).
→
If $\vec{\text{a}} , \vec{\text{b}} , \vec{\text{c}}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}$ is equally inclined to $\vec{\text{a}} , \vec{\text{b}}$ and $\vec{\text{c}}$ Also, find the angle which $\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}$ makes with $\vec{\text{a}} \text{ or }\vec{\text{b}} \text{ or } \vec{\text{c}}.$
→
If f(x) = x3 + ax2 + bx + c has a maximum at x = -1 and minimum at x = 3. Determine a, b and c.
→
Suppose that 6% of the people with blood group O are left handed and 10% of those with other blood groups are left handed 30% of the people have blood group O. If a left handed person is selected at random, what is the probability that he/she will have blood group O?
→
Evaluate the following integrals:

$\int\sin^{-1}\sqrt{\text{x}}\text{dx}$

→
Solve the following systems of homogeneous linear equations by matrix method:
2x - y + 2z = 0
5x + 3y - z = 0
x + 5y - 5z = 0
→
Two groups are competing for the positions of the Board of Directors of a Corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
→