# 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 Mar 31, 2019 at 3:41
• But what about others options, how can I check? Mar 31, 2019 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. Mar 31, 2019 at 4:45
• Making any progress? Apr 1, 2019 at 12:18
• Not yet but I'm trying. Apr 1, 2019 at 14:31

For option $$\bf {(1)}$$ : If the solution is $$(C_1+1)y_1+(C_2-C_1)y_2-C_2y_3$$Sum of coefficients is $$(C_1+1)+(C_2-C_1)-C_2=1$$Hence option $$(1)$$ is correct.
For option $$\bf {(2)}$$ : If the solution is $$C_1(y_1-y_2)+C_2(y_2-y_3)=C_1y_1+(C_2-C_1)y_2-C_2y_3$$ Sum of coefficients is $$C_1+(C_2-C_1)-C_2=0$$Hence option $$(2)$$ is incorrect.
For option $$\bf {(3)}$$ : If the solution is $$C_1(y_1-y_2)+C_2(y_2-y_3)+C_3(y_3-y_1)=(C_1-C_3)y_1+(C_2-C_1)y_2+(C_3-C_2)y_3$$Sum of coefficients is $$C_1-C_3+C_2-C_1+C_3-C_2=0$$Hence option $$(3)$$ is again incorrect.
For option $$\bf {(4)}$$ : If the solution is $$C_1(y_1-y_3)+C_2(y_3-y_2)+y_1=(C_1+1)y_1+(-C_2)y_2+(C_2-C_1)y_3$$Sum of coefficients is $$C_1+1-C_2+C_2-C_1=1$$Hence option $$(4)$$ is correct.