Solve the differential equation (x^2 – yx^2)dy + (y^2 + xy^2)dx =… - Vidyadip

Question

Solve the differential equation $(x^2 – yx^2)dy + (y^2 + xy^2)dx = 0$

✓

Answer

$\left( x ^2- y x ^2\right) d y +\left( y ^2+ xy y ^2\right) d x =0$
$\therefore x ^2(1- y ) d y + y ^2(1+ x ) d x =0$
$\therefore x ^2(1- y ) d y =- y ^2(1+ x ) d x$
$\therefore\left(\frac{1-y}{y^2}\right) d y=-\left(\frac{1+x}{x^2}\right) d x$
$\operatorname{lntegrating~on~both~sides,~we~get~}$
$\int\left(\frac{1-y}{y^2}\right) d y=-\int\left(\frac{1+x}{x^2}\right) d x$
$\therefore \int \frac{1}{y^2} d y-\int \frac{1}{y} d y=-\int \frac{1}{x^2} d x-\int \frac{1}{x} d x$
$\therefore \frac{y^{-1}}{-1}-\log |y|=\left(\frac{x^{-1}}{-1}\right)-\log |x|+ c$
$\therefore-\frac{1}{y}-\log |y|=\frac{1}{x}-\log |x|+ c$
$\therefore \log | x |-\log | y |=\frac{1}{x}+\frac{1}{y}+ 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 "Solve the Following Question.(3 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

A toy manufacturing company produces five types of toys. Each toy has to go through three machines A, B, C in the order ABC. The time required in hours for each process is given in the following table.

Type	1	2	3	4	5
Machine A	16	20	12	14	22
Machine B	10	12	4	6	8
Machine C	8	18	16	12	10

Solve the problem for minimizing the total elapsed time.

Evalute : $\int \frac{x^3}{16 x^8-25} d x$

Find area of the region bounded by the parabola $x^2 = 4y$, the Y-axis lying in the first quadrant and the lines $y = 3$

Using the rules of negation, write the negations of the following:
(i) (p → r) ∧ q
(ii) ~(p ∨ q) → r
(iii) (~p ∧ q) ∧ (~q ∨ ~r)

For bivariate data.
$
\bar{x}=53, \bar{x}=28, b_{y x}=-1.2, b_{x y}=-0.3
$
Find,
(i) Correlation coefficient between $\mathrm{X}$ and $\mathrm{Y}$.
(ii) Estimate $Y$ for $X=50$
(iii) Estimate $X$ for $Y=25$

The difference between true discount and banker’s discount on 6 months hence at 4% p.a. is $\text{₹}$ 80. Find the true discount, banker’s discount, and amount of the bill.

4,000 bedsheets worth ₹ 6,40,000 were insured for $\left(\frac{3}{7}\right)^{t h}$ of their value. Some of the bedsheets were damaged in the rainy season and were reduced to $40 \%$ of their value. If the amount recovered against damage was ₹ 32,000. Find the number of damaged bedsheets.

If $A=\left[a_{i j}\right]_{3 \times 3}$ where $a_{i j}=2(i-j)$. Find $A$ and $A^{\top}$. State whether $A$ and $A^{\top}$ both are symmetric or skew-symmetric matrices.

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes

A person plans to put $\text{₹}$ $400$ at the beginning of each year for $2$ years in a deposit that gives interest at $2 \%$ p.a. compounded annually. Find the amount that will be accumulated at the end of $2$ years. [Given: $(1.02)^2=1.0404$ ]