# How to get the general solutions?

Three solutions of a certain second order non-homogeneous linear differential equation are $$y_1(x)=1+xe^{x^2} , y_2(x)= (1+x)e^{x^2}-1 , y_3(x)=1+e^{x^2}$$ Which of the following are general solutions of the differential equation ?

1.$$(C_1+1)y_1+(C_2-C_1)y_2-C_2y_3.$$

2.$$C_1(y_1-y_2)+C_2(y_2-y_3).$$

3.$$C_1(y_1-y_2)+C_2(y_2-y_3)+C_3(y_3-y_1).$$

4.$$C_1(y_1-y_3)+C_2(y_3-y_2)+y_1$$ , Where $$C_1 , C_2$$ and $$C_3$$ are arbitrary constant.

My efforts is that , option third will be wrong (since the second order differential equation is given, So in the general solution , the number of constant should be two only.)

• I'd be tempted to agree with you – George Dewhirst Mar 31 at 3:41
• But what about others options, how can I check? – user118413 Mar 31 at 4:18
• I would check to see whether there are (constant) values of $C_1$ and $C_2$ that allow you to recover each of $y_1,y_2,y_3$ as a solution. – Gerry Myerson Mar 31 at 4:45
• Making any progress? – Gerry Myerson Apr 1 at 12:18
• Not yet but I'm trying. – user118413 Apr 1 at 14:31