Question

Consider a second-order linear homogeneous equation

Suppose that are two solutions. Show that $y_{3}=y_1+3y_2$ is also a solution to the equation (plug it in and use the fact that $y_1$ and $y_2$ are solutions).

Answer 1

11 LO Consider a second-order linear homogeneous equation y'+ Plt) y'+914) Y=0 Suppose Y, ,9, are cre two solution of then yı' + plk) y + 914) , 50 and y" + pidy,' +90+%, =0 2 1) Now, ym + plu) Yz +9fEY, 3 1 ) = (9, +393) + ÞI+) 19,+34) + %,!+(9:134) y' +39) + b1E) (9' +39,') + 9,4) (9,434) U a = (y + pledg,'t +3 pl E-) y,' + 39,05) 92 11 + Þix) y; +9,1F) y, ) +3 ( y + $1E)y,!+ 9411-) 92 qp1>) 92) 0+310) . - 11 w) 93.+ pla) yz + 91699; =O ..z is a à solution to o