Given a second order linear homogeneous differential equationa2(x)y''+a1(x)y'+a0(x)y=0we know that a fundamental set for this ODE consists of a pair linearly independent solutions y1,y2. But there are times when only one function, callity1,is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order.First, under the necessary assumption the a2(x)0we rewrite the equation asy''+p(x)y'+q(x)y=0,p(x)=a1(x)a2(x),q(x)=a0(x)a2(x)Then the method of reduction of order gives a second linearly independent solution asy2(x)=Cy1u=Cy1(x)e-p(x)dxy12(x)dxwhere Cisan arbitrary constant. We can choose the arbitrary constant tobe anything we like. One useful choice isto choose Cso that all theconstants in front reduce to1. For example, ifwe obtain y2=C3e2x then we can choose C=13so that y2=e2x.Given the problemy''-4y'+29y=0and a solution y1=e2xsin(5x)Applying the reduction of order method we obtain the followingy12(x)=p(x)= and e-p(x)dx=Sowe havee-p(x)dxy12(x)dx=,dx=Sowe havee-p(x)dxy12(x)dx=,dx=Finally, after making a selection of a value for Cas described above (you have to choose some nonzero numerical value)we arrive aty2(x)=Cy1u=So the general solution toy''-5y'+4y=0 can be written asy=c1y1+c2y2=c1,+c2