Question

If y1 = 2x+2 and y2 = -x^2/2 are the solutions of the equation y= xy' +( y')^2/2 then are the constant multipliers c1 , c2 are arbitrary, also the solutions of the given D.E. ? Is the sum y1+ y2 a solution ? Justify your answer.

Answer 1

Let

$$y=c_1y_1+c_2y_2$$ $$y=c_1(2x+2)+c_2(-{x^2 \over 2})$$

Then

$$y'=2c_1-c_2x$$

$$c_1(2x+2)+c_2(-{x^2 \over 2})=x(2c_1x-c_2x)+{(2c_1-c_2x)^2 \over2}$$

$$({1\over 2}c_2-{1\over 2}c_2)x^2+(2c_1-2c_1+2c_1c_2)x+2c_1-2{c_1}^2=0$$

$$2c_1c_2x+2c_1-2{c_1}^2=0$$

We see that constant multipliers c₁ and c₂ are not arbitrary. We have to satisfy

$$c_1c_2=0, c_1-{c_1}^2=0$$

Therefore, the sum y₁+y₂ is not the solution of the equation.