Question

Please solve all three. Thank you very much

5. (a) Let a be a constant (we can write “a ER” to mean “a is a real number”). Verify that y(x) = ci cos(ax) + C2 sin(ax) is a solution for y" = -a’y, where C1,C2 ER. (b) Consider the hyperbolic trigonometric functions defined by cosh(x) = et tex 2 ex – e- sinh(x) = * d Show that I cosh(x) = sinh(x) and sinh(x) = cosh(x). (e) Verify that y(x) = ci cosh(ax) + ca sinh(ax) is a solution for y" = a’y, C1, C2 E R.

Answer 1

CA dil 5) a) You cicos cax) + a sin car) wheze ence etR of a constane Y'(a) - (464)= d facos Ca24) +s_sincand)) - cos(az) C Casio (2) y!cous-csopcan): at ce Bos (an).a chy chain rule of derivative - cax) – 2. dla f'cas) 1- Y"60)=d you d (agasan case) i cea cascass) de faasin.cam) +di Caracossacx)) E-Giacos Cax).a - Gasin caxla - rac cos Camc) - Ga sin cave) ca? ( cu coscam) sin colou) Japhene -any coe) s'. Y c) Satisfies yhle-ay => Y60) is Solo of yaraly. b considez hyperbolic tzianometzic functions defined by , cash.com eztése and sinhwa- equiem Now, I (coshal - Corte = Letter Peter Brue- sinha 2

Hence, tal cosha)= sinhx – Now, & (sinha) - (ex-pay Les-Lē%E0 2 = cosha Hence, & (sinhre) - cosha c) y(x) = a coshax) +G sinh cax) . t' ()- sinh cao).0 + Cu(Osh(ax).a - fby chain zule of dezivative n a flan) a ficax) ) Y'ac) a ci sinh camel tace cosh came) y accosh (ax).atacsinh cave)a - a² cash case) + a² c sinhcan) ( c osh ca) + ( Sinh (3) Y"() - Byee Hence, y(x) satisfies all a q²y - Y Cal) is solution of y-a2y.