Let c₁, c₂, c₃, c₄ be arbitrary constants. The order of the differential equation, corresponding to y=c₁ e^x+ c ₂ e ^ _ e x + c ₃ ^2 x - c ₄ ( ^2 x -1 ) is

Step-by-step solution

y=c₁ e^x+c₂ e^ c x +c₃ ^2 x-c₄ ( ^2 x-1 ) y=c₁ e^x+c₂ x+ (c₃+c₄ ) ^2 x y=c₁ e^x+c₂ x+c₃^ ^2 x ...(i) where c₃^ =c₃+c₄y^ =c₁ e^x+c₂+c₃^ 2 x ...(ii) y^ =c₁ e^x+c₂ 0+2 c₃^ 2 x ...(iii) y^ =c₁ e^x+c₂ 0+c₃^ (-4 2 x) ...(iv) Eliminating c₁, c₂ & c₃ from equation (i), (ii), (iii) & (iv), we get | array cccc y & -e^x & -x & - ^2 x y^ & -e^x & -1 & - 2 x y^ & -e^x & 0 & -2 2 x y^ & -e^x & 0 & +4 2 x array |=0 Clearly, after solving above determinant, we will get a differential equation with order 3 .