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.

Answer

`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`.