## Question

Show that y_1=1 and y_2=ln(x) are both solutions of the non-linear differential equation y''+(y')^2=0.

1. Is y_1+y_2 also a solution? Demonstrate.
2. Is c_1y_1+c_2y_2 also a solution? Demonstrate.

y''+(y')^2=0.

y_1=1, y_2=ln(x).

y_1=1 is solution because 1''+(1')^2=0+0^2=0.

y_2=ln(x) is solution because (ln(x))''+((ln(x))')^2=(1/x)'+(1/x)^2=-1/x^2+1/x^2=0.

1. y_1+y_2=1+ln(x) is also a solution because

(1+ln(x))''+((1+ln(x))')^2=(0+1/x)'+(0+1/x)^2=

=(1/x)'+(1/x)^2=-1/x^2+1/x^2=0.

2. c_1y_1+c_2y_2=c_1+c_2ln(x) is not a solution because

(c_1+c_2ln(x))''+((c_1+c_2ln(x))')^2=(0+c_2/x)+(0+c_2/x)^2=

=c_2(1/x)'+(c_2/x)^2=-c_2/x^2+c_2^2/x^2=c_2/x^2(c_2-1)!=0.