ญ 4. Consider the second ordinary differential equation y" = (2x-5y)y'. (a) Derive the determining equations for the symmetries of equation (2). (b) Solve the determining equations and find the symmetries of (2). 1. 2. -2x- მ2 მუ მუ + 5y- + 5n — 2č - მx2 მყ მყ მč 2x მყ - 5y მყ - θε +2 მ2 მყ მთმყ მ-n მ/2 2 3. მ -2x + 5ყ მთ 4. მმყ = მუ + მუ = 0. მე მ2 მყ2 = = 0. = 0.

Consider the differential equation y'= et. (a) Draw its direction field. ? A. y C. 1.1. 10 T (b) Find a general solution to the differential equation help (formulas) (c) Draw a solution on top of the direction field. ? î J.J. į į jį į ]]] J.J.J. TIT i B. D. 111 TT 11111 1 1 1 1 TTT +-+ 1 1 1 1 TT TTT 1 J J J J J 1 T 1 J J T TTT -10| 111 1 T 1 1 1 } } TTT N TTTTTTTT 11 TT TT 1 11 1 ] J -16 JILL IIIII | | | | | TILLI 1 ÿ 1 1 y 1. 1 LI 11 10 1 111 11 LII 111 11111 1. J.J. III]. [|]] |||

Determine the constant ? such that the differential equation(??^2? + 2?^2)?? + (?^3+ 4??)?? = 0is exact and solve the resulting exact equation.

Q4 (A): Derive the differential equation governing the buckling of the column shown in figure knowing that the column has a uniform circular cross section, then determine the first buckling load? P y L (x,y) TTTT