Download App

Differential Equations — Maths STD 12 Science — Question

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

Answer

Here, $\big(\text{y}^2-2\text{xy}\big)\text{dx}=\big(\text{x}^2-2\text{xy}\big)\text{dy}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{y}^2-2\text{xy}}{\text{x}^2-2\text{xy}}$
It is a homogeneous equation.
Put y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So,
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2\text{x}^2-2\text{xvx}}{\text{x}^2-2\text{xvx}}$
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2-2\text{v}}{1-2\text{v}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{\text{v}^2-2\text{v}}{1-2\text{v}}-\text{v}$
$=\frac{\text{v}^2-2\text{v}-\text{v}+2\text{v}^2}{1-2\text{v}}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\frac{3\text{v}^2-3\text{v}}{1-2\text{v}}$
$\frac{1-2\text{v}}{3(\text{v}^2-\text{v})}\text{dv}=\frac{\text{dx}}{\text{x}}$
$\frac{-(2\text{v}-1)}{3(\text{v}^2-\text{v})}\text{dv}=\frac{\text{dx}}{\text{x}}$
$\int\frac{2\text{v}-1}{\text{v}^2-\text{v}}\text{dv}=-3\int\frac{\text{dx}}{\text{x}}$
$\log\big|\text{v}^2-\text{v}\big|-3\log|\text{x}|+\log\text{C}$
$\text{v}^2-\text{v}=\frac{\text{C}}{\text{x}^3}$
$\frac{\text{y}^2}{\text{x}^2}-\frac{\text{y}}{\text{x}}=\frac{\text{C}}{\text{x}^3}$
$\text{y}^2-\text{xy}=\frac{\text{C}}{\text{x}}$
$\text{x}\big(\text{y}^2-\text{xy}\big)=\text{C}$

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

Using differentials, find the approximate values of the following:
$\sqrt{49.5}$
→
Verify the Rolle’s theorem for each of the functions:
$\text{f(x)}=\sin^4\text{x}+\cos^4\text{x}\text{ in }\Big[0,\frac{\pi}{2}\Big].$
→
Evaluate the definite integral $\int _ { 0 } ^ { 1 } x e ^ { x ^ { 2 } } d x.$
→
Find a particular solution of the differential equation $(\text{x}-\text{y})(\text{dx}+\text{dy})=\text{dx}-\text{dy},\ \text{given that y}=-1,$ $\text{when x}=0. \ (\text{Hint: put x}-\text{y}=\text{t})$
→
If $\triangle=\begin{vmatrix}1&\text{x}&\text{x}^2\\1&\text{y}&\text{y}^2\\1&\text{z}&\text{z}^2\end{vmatrix},$ $\triangle_1=\begin{vmatrix}1&1&1\\\text{yz}&\text{zx}&\text{xy}\\\text{x}&\text{y}&\text{z}\end{vmatrix},$ then prove that $\triangle+\triangle_1=0$
→
Differentiate the following functions with respect to x:
$\log\sqrt{\frac{\text{x}-1}{\text{x}+1}}$
→
Construct $a_{2 \times 2}$​​​​​​​ matrix, where,
  1. $\text{a}_\text{ij}=\frac{(\text{i}-2\text{j})^2}{2}$
  2. $\text{a}_\text{ij}=|-2\text{i}+3\text{j}|$
→
If lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} \text{and} \frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then find the value of k and hence find the equation of the plane containing these lines.
→
Solve the following systems of linear equations by cramer's rule:
2x - 3y - 4z = 29,
-2x + 5y - z = -15,
3x - y + 5z = -11
→
Solve the following LPP graphically:
Maximise Z = 105x + 90y
subject to the constraints
x + y $\leq$ 50
2x + y $\leq$80
x $\geq$ 0, y $\geq$ 0.
→