Question

13.)

Use the method for solving homogeneous equations to solve the following differential equation (x2 + y2) dx + Swy dy=0 C, where C is an arbitrary constant Ignoring lost solutions, if any, an implicit solution in the form Fixy)-Cis (Type an expression using x and y as the variables.)

Answer 1

Given (x2+43) da + gxy dy-0 Rewolning above equations -(22+43) - dy = 'gay dat pret Yaval in o → om avtale me lejuation o gines » - -(x²+(12²) V tem gacowi -(x2+v°) V4 x V - gva? -%2C1tv2) V + 2 dv = V + 2 dv 9Vx - (1+12) V av V -CIV 9V 12 -(1+12) IV - Citlov3)

> gv dv -- I da 1+10v2 This equation is in variable separable form ... Integrating on both sides Sien oz tv -- Sz toe +lege Bag C.-constant → log C1+10v2) = - log x + logo loq C1 + 10v?)" = -logxt logo → dogCl + 4.0 V²) 120 + loga = logc = ► log [ x C1+10023460] 2017100239/20 = logo -

Resubstprerje y ra ieva Use I'above equation = x 1 +10 - - = c. 2? = « [1092 place ► gho [2271042.jº20 = C Ratsing powere 20 on both sides x²[x²+10y 279 = (20 2+[x2+10y+jº = c' Thus an implicit solupon in the form fcxiyle cis a>[22+10 y?J° = C e' is an arbitrary constant