Chapter 4.2, Problem 16P

Consider the differential operator $T$ defined by $T(y)=yy\text{'}\text{'}+{\left(y\text{'}\right)}^2$. Show that $T(y)=0$ is a nonlinear equation by three methods below.

(a) Explain why $T(y)=0$ cannot be put in the form of linear equation $L\left(y\right)=y\text{'}\text{'}+py\text{'}+qy=0$.

(b) Show that $T$ fails to satisfy the definition of a linear operator and therefore the equation $T(y)=0$ is a nonlinear equation.

(c) Verify that $y_1\left(t\right)=1,y_1\left(t\right)=t^{1/2}$ are solutions of the differential equation $T(y)=0$ for $t>0$, but $C_1+C_2{t}^{1/2}$ is not, in general, a solution of this equation.