Question

One solution of the differential equation y" + y = 0 is yı = cosx. Use the method of reduction of order (let y = uy), a se Select the correct answer. Submit your work for full credit. a. y = cost y=xcosx b. y = sinx C. y = ef d. y=e- e.

Answer 1

yllty=o. Given that y = cosse is one solution of above equation To find another solution. Let y=u. Y = u.coss y= u. cosx. : y = e. -sins & u. coss - y = - u cosx-ul sige - ul.sinx + ul. cose y = -u.cosa - 2.4. sinx fui COX we get - ul. cosse u-vi put these values in equation y"+y=0 all cosa zu sinn - u. cosa tu cour=0 zu' sinaco Put u=v vi cosa 2 Sinx.v=0 2. tana.v=0 which is reduced first ordes equation in the form VI + PO.V = que, where poe) = -2.tana, 902) = 0 . Its solution is given by, Spises da v. e Sortem = sacen. Spende da to du So faranx de tc. V. -2. lo sean - С em sees -C V secz = C V= c secer

u = c. seca e usu) I c. ser²x dx + d in c-tanx &d .. Y₂ = (c. tanxtd). coss c. tann.cosxt d. cors - C- sine cour & d corne cose 42= e sin t d. cosse But cosa is already second solution is is solution. sinn. . y=sinn is required option.