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   This differential equations is called Fowler's second order differential (non-linear) equation.  The solution is not found in closed expression of known simple functions.  We can find the answer by using Taylor series expansion for y(x).

           Solving    d²y/dx² = - x² y    --- (1)

  When we differentiate y(x) wrt x twice, the exponent of x reduces by 2 in y".  On the RHS of (1) we have x² y, so exponent is increased by 2.  Equation (1) will be valid if the exponent of x of  (n+1)th term = 4 + exponent of n th term. 

We can find two linearly independent solutions as:

y_1(x)=a_0+a_1 x^4+a_2 x^8+ a_3 x^{12} +a_4 x^{16}+a_5 x^{20}+..--(2)\\\\y_1'(x)=4a_1 x^3+8 a_2 x^7+12 a_3 x^{11}+ 16 a_4 x^{15}+20 a_5 x^{19}+..\\\\y_1''(x)=4*3a_1 x^2+8*7a_2 x^6+12*11a_3 x^{10}+ 16*15 a_4 x^{14}+..--(3)\\\\-x^2 y_1(x)=-(a_0x^2+a_1 x^6+a_2 x^{10}+ a_3 x^{14} +a_4 x^{18}+.. )..(4)\\\\Equating\ (3)\ and\ (4)\\\\a_1=-\frac{a_0}{4*3},\ a_2=-\frac{a_1}{8*7}=\frac{a_0}{4*3*8*7},\ a_3=-\frac{a_2}{12*11} =\frac{a_0}{4*3*8*7*12*11}..

Similarly for another independent solution : 

y_2(x)=b_0 x^1+ b_1 x^5+b_2 x^9 + b_3 x^{13}+b_4 x^{17}+..--(5)\\\\y_2''(x)=5*4b_1x^3+9*8b_2 x^7+13*12 b_3 x^{11}+...(6)\\\\-x^2 y_2(x)=-(b_0 x^3+b_1 x^7+b_2 x^{11}+b_3 x^{15}+..)..--(7)\\\\equating\ (6)\ and\ (7)\ we\ get\\\\b_1=-\frac{b_0}{5*4},\ b_2=-\frac{b_1}{9*8}=\frac{b_0}{9*8*5*4},\ b_3=-\frac{b_2}{13*12}=-\frac{b_0}{13*12*9*8*5*4},..

Now we have the final general solution as :

y_1(x)=a_0*[1-\frac{x^4}{12}+\frac{x^8}{12*56}-\frac{x^{12}}{12*56*132}+...]\\\\y_2(x)=b_0*[ x-\frac{x^5}{5*4}+\frac{x^9}{9*8*5*4}-\frac{x^{13}}{13*12*9*8*5*4} +.. ]\\\\y(x)=c_1\ y_1(x)+c_2\ y_2(x),\ \ ---(8)

here c1 and c2 are real constants.

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