1 solve the differential equation x2dy +(xy+y2)dx=0 given y=1, when x=1 2 find the particular solution of the - Maths - Differential Equations

Question

1 solve the differential equation x2
​dy +(xy+y2)dx=0 given y=1, when
x=1
2 find the particular solution of the differential equation xdy/dx
+ y - x + xycotx=0 , given that when x=^/2, y=0

Varun.Rawat · Accepted Answer

(1) $x^{2} d y + (x y + y^{2}) d x = 0 â‡’ x^{2} d y = âˆ’ (x y + y^{2}) d x â‡’ \frac{d y}{d x} = \frac{âˆ’ (x y + y^{2})}{x^{2}}$ Let y = vx $âˆ´ \frac{d y}{d x} = v + x \frac{d v}{d x}$ The differential equation can be written as $v + x \frac{d v}{d x} = \frac{âˆ’ (x^{2} v + v^{2} x^{2}}{x^{2}}) â‡’ x \frac{d v}{d x} = âˆ’ (2 v + v^{2}) âˆ´ âˆ« \frac{1}{(2 v + v^{2})} â\dots† v = âˆ’ âˆ« \frac{1}{x} â\dots† x â‡’ âˆ« (\frac{1}{2 v} âˆ’ \frac{1}{2 (v + 2)}) â\dots† v = âˆ’ âˆ« \frac{1}{x} â\dots† x â‡’ \frac{1}{2} âˆ« \frac{1}{v} â\dots† v âˆ’ \frac{1}{2} âˆ« \frac{1}{v + 2} â\dots† v = âˆ’ âˆ« \frac{1}{x} â\dots† x â‡’ \frac{1}{2} \log â¡ | v | âˆ’ \frac{1}{2} \log â¡ | v + 2 | = âˆ’ \log â¡ | x | + \log â¡ C â‡’ \frac{1}{2} \log â¡ | \frac{v}{v + 2} | = \log â¡ \frac{C}{x} â‡’ \log â¡ | \frac{\frac{y}{x}}{\frac{y}{x} + 2} | = 2 \log â¡ \frac{C}{x} â‡’ \log | \frac{y}{y + 2 x} | = 2 \log â¡ \frac{C}{x}$ log y2x+y = logC2x2⇒C2 = x2y2x+y 1Put x = y = 1 in 1, we getC2 = 12+1 = 13⇒C = 13Now, from 1, we get 13 = x2y2x+y⇒2x+y-3x2y = 0 Kindly ask the remaining query in different thread

1.solve the differential equation x2 ​dy +(xy+y2)dx=0 given y=1, when x=1. 2. find the particular solution of the differential equation xdy/dx + y - x + xycotx=0 , given that when x=^/2, y=0.

1.solve the differential equation x² dy +(xy+y²)dx=0 given y=1, when x=1. 2. find the particular solution of the differential equation xdy/dx + y - x + xycotx=0 , given that when x=^/2, y=0.