Question

please show all steps

4. Some equations that are not separable can be made separable by an appropriate substitution. Differential equations of the form y = f (!) are called Euler-homogeneous. These can be solved by letting v = y/t or y = ut. Using the product rule, dy du di = v + tai so that the differential equation becomes du v+t du f(0) - 0 dt t Use this technique to solve dy 2y+ + +4 - Leave your answer in implicit form.

Answer 1

given that 44 +4 tys Differential equation Solve the above Solution - from o dt ty Aboue Differential ean is homogeneous first ordeo Differential equation put y = ut - olen Se 10+) dite u but do not oo from egno **, we have from eq? @ (+du, 2(6+)4 +++ 6+ it + (6+)3 utt do = 204+4+44 u3+4 e d+ 204+1 203 - L 2L04+1-14 U3 63 64 + 1 tdu - Su 13 -do- 4+1 home do = { dt sites du = 1 200 + An(c) Integrating ult=2 443d0=d2 widus toz *+1) - lo (t) tincc) In (441) V3 = b(cth

Take ontilag we have (64+1jY4 = ct 64+1 = c4+4 - sinu ylt) = twit) . Then from @qn 3 probablement (glt)) #1 = citt 441t) +4 = c +8 741) + * 8 74EC