Question

Solve the following differential equations :

(i) (e^y + 1) cos x dx + e^y sin x dy = 0.

(ii) (1 + x²) dy = (1 + y²) dx.

Answer 1

(i) (e^y + 1) cos x dx + e^y sin x dy = 0.

Given equation is

(ii) (1 + x²) dy = (1 + y²) dx.

Given

Answer 2

1)

2)

(1+x2)dy=(1+y2)dx(1+x2)dy=(1+y2)dx

dy1+y2=dx1+x2dy1+y2=dx1+x2

∫dy1+y2=∫dx1+x2∫dy1+y2=∫dx1+x2

tan−1y=tan−1x+C1tan−1⁡y=tan−1⁡x+C1

tan−1y−tan−1x=C1tan−1⁡y−tan−1⁡x=C1

tan−1(y−x1+xy)=C1tan−1⁡(y−x1+xy)=C1

y−x1+xy=tanC1y−x1+xy=tan⁡C1

y−x1+xy=C2y−x1+xy=C2

y−x=C2(1+xy)y−x=C2(1+xy)

x−y+C(1+xy)=0