If I have differential equation of fourth order with four solutions $\alpha _1,\alpha _2,-\alpha _1,-\alpha _2 $ where we have appearing the solutions in the form $e^{\text{$i\alpha $}_1 x},e^{\text{$i\alpha $}_2 x},e^{-\text{$i\alpha $}_1 x},e^{-\text{$i\alpha $}_2 x}$. Is it possible to force Mathematica to reduce order of this equation on 2 but to keep two positive solutions $ \alpha _1,\alpha _2 $ which should be solutions of this equation
DSolve[Y''''[x] + a1 Y''[x] + a2*Y[x] == 0, Y[x], x]
output
$$ Y(x)\to c_1 e^{\frac{x \sqrt{-\sqrt{\text{a1}^2-4 \text{a2}}-\text{a1}}}{\sqrt{2}}}+c_2 e^{-\frac{x \sqrt{-\sqrt{\text{a1}^2-4 \text{a2}}-\text{a1}}}{\sqrt{2}}}+c_3 e^{\frac{x \sqrt{\sqrt{\text{a1}^2-4 \text{a2}}-\text{a1}}}{\sqrt{2}}}+c_4 e^{-\frac{x \sqrt{\sqrt{\text{a1}^2-4 \text{a2}}-\text{a1}}}{\sqrt{2}}} $$
So solutions are now in the form $e^{\text{$i\beta $}_1 x},e^{-\text{$i\beta $}_1 x}$
Now I want $ \alpha _1,\alpha _2 $ to be equal with two solutions of reduced equation ${\text{$\beta $}_1 },{-\text{$\beta $}_1 }$ on second order which has solutions $\sqrt{-\text{b1}-\text{b2}}$ and $-\sqrt{-\text{b1}-\text{b2}}$. Or to make other possible combination but to neglect two of them, maybe other combination $ ({+/-\alpha _i,+/-\alpha _j})$ where $ i,j=1,2. $ Reduced equation
DSolve[Y''[x] + b1 Y[x] + b2*Y[x] == 0, Y[x], x]
$$ Y(x)\to c_1 e^{x \sqrt{-\text{b1}-\text{b2}}}+c_2 e^{x \left(-\sqrt{-\text{b1}-\text{b2}}\right)} $$
Maybe idea can be first to extract solutions of first equation and then make conditions for second. It is good if I can keep two positive solutions from first equation. Maybe Mathematica has some functions for that or any help can be good.
Atention: Constants C1,... can be different.
