ODE No. 1214

$$y(x)(-a^2+x^2(2a+2n+1)+a(-1{)}^n-x^4)+x^2{y}^{″}(x)=0$$ ✓ Mathematica : cpu = 0.200588 (sec), leaf count = 260

DSolve[((-1)^n*a - a^2 + (1 + 2*a + 2*n)*x^2 - x^4)*y[x] + x^2*Derivative[2][y][x] == 0,y[x],x]
 

$$\left\{\{y(x)\to \frac{c_1{e}^{-\frac{x^2}{2}}{2}^{\frac{1}{4}(\sqrt{4a^2-4a(-1{)}^n+1}+2)}{\left(x^2\right)}^{\frac{1}{4}(\sqrt{4a^2-4a(-1{)}^n+1}+2)}U(\frac{1}{4}(-2a-2n+\sqrt{4a^2-4(-1{)}^{n}a+1}+1),\frac{1}{2}(\sqrt{4a^2-4(-1{)}^{n}a+1}+2),x^2)}{\sqrt{x}}+\frac{c_2{e}^{-\frac{x^2}{2}}{2}^{\frac{1}{4}(\sqrt{4a^2-4a(-1{)}^n+1}+2)}{\left(x^2\right)}^{\frac{1}{4}(\sqrt{4a^2-4a(-1{)}^n+1}+2)}{L}_{\frac{1}{4}(2a+2n-\sqrt{4a^2-4(-1{)}^{n}a+1}-1)}^{\frac{1}{2}(\sqrt{4a^2-4(-1{)}^{n}a+1}+2)-1}\left(x^2\right)}{\sqrt{x}}\}\right\}$$ ✓ Maple : cpu = 0.156 (sec), leaf count = 71

dsolve(x^2*diff(diff(y(x),x),x)+(-x^4+(2*n+2*a+1)*x^2+(-1)^n*a-a^2)*y(x)=0,y(x))
 

$$y\left(x\right)=\frac{\mathrm{WhittakerW}(\frac{n}{2}+\frac{a}{2}+\frac{1}{4},\frac{\sqrt{1-4{(-1)}^{n}a+4a^2}}{4},x^2)c_2+\mathrm{WhittakerM}(\frac{n}{2}+\frac{a}{2}+\frac{1}{4},\frac{\sqrt{1-4{(-1)}^{n}a+4a^2}}{4},x^2)c_1}{\sqrt{x}}$$