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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.)

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  • $\begingroup$ I'd be tempted to agree with you $\endgroup$
    – fGDu94
    Mar 31, 2019 at 3:41
  • $\begingroup$ But what about others options, how can I check? $\endgroup$
    – user118413
    Mar 31, 2019 at 4:18
  • $\begingroup$ 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. $\endgroup$ Mar 31, 2019 at 4:45
  • $\begingroup$ Making any progress? $\endgroup$ Apr 1, 2019 at 12:18
  • $\begingroup$ Not yet but I'm trying. $\endgroup$
    – user118413
    Apr 1, 2019 at 14:31

1 Answer 1

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Note: Any linear combination of solutions of a non-homogeneous second order linear differential equation is again a solution if the sum of the coefficients is one.

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.

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