Question

Classification of ODEs Determine the order and linearity/nonlinearity of the following scalar ODEs (a) (10 points) y"(t) sin(t) y(t) +t2 (b) (10 points) y'(t)-sin(y(t))- t Note: If the equation is linear, vou need to prove it by the linearity property. If the equation is nonlinear, only determine the term in the equation that makes it nonlinear S. S.

Answer 1

Y^'' = sin t y(t) + t^2 order of the differential equation highest derivative in the differential equation. Order of the differential equation is '2'. Order is '2' linearity means. Y = y_1 + y_2 + y_3 + .... If y_L = y_1 + y_2 in given differential equation y = y_1 y_1^'' = sin t y_1(t) + t^2 equation (1) y = y_2 y = y_2 y_2^'' = sin t y_2(t) + t^2 equation (2) add both equations y_1^'' + y_2^'' = sin t y_1(t) + sin t y_2(t) + t^2 + t^2 y_1^'' + y_2^'' = sin t(y(t) + y_2(t)) + 2t^2 y_2(t) = sin t(y(t)) + 2t^2 y(t) notequalto y_L(t) so this is not linear differential equation. non linear differential equation. Y^'(t) = sin(y(t)) - t the order A the differential equation is 'f' \r\n y^'(t) = sin(y(t)) - t y_1^'(t) = sin(y_1(t)) - t equation (1) y_2^'(t) = sin(y_2(t)) - t equation (2) y_1^'(t) + y_2(t) = sin(y_1(t)) + sin(y_2(t)) \r\n