1. Show that y1=cosx and y2=sinx are solution of the DE y''+y=0.verify that ⱷ(x)=c1cosx+c2sinx is also a solution of the DE.
2. Given that y1 and y2 are the solution of the DE y''+y^2=0. Is y3 =c1y1+c2y2 also a solution of the DE?
1. Show that y1=cosx and y2=sinx are solution of the DE y''+y=0.verify that ⱷ(x)=c1cosx+c2sinx is also a solution of the DE.
2. Given that y1 and y2 are the solution of the DE y''+y^2=0. Is y3 =c1y1+c2y2 also a solution of the DE?
Just to let the helpers here know, this question has also been posted at MMF, and I have offered hints on how to proceed.
For the first one, you will need to find $\displaystyle \varphi''(x)$ and substitute into the given ODE.
For the second one, you are told:
$\displaystyle y_1''+y_1^2=0$
$\displaystyle y_2''+y_2^2=0$
Multiply the first equation by $\displaystyle c_1$ and the second by $\displaystyle c_2$, then add the equations. What do you find?
If $\displaystyle y_3\equiv c_1y_1+c_2y_2$ is a solution, then we must have:
$\displaystyle y_3''+y_3^2=0$
Substitute for $\displaystyle y_3$ into the ODE using the above definition, and see if this is equal to your earlier result...