Chapter 8 - Linear Differential Equations of Order n - 8.1 General Theory for Linear Differential Equations - Problems - Page 503: 19

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Given: $y''+x^2y+e^xy=0$ with $y(0)=0\\ y'(0)=0$ From $y''+x^2y+e^xy=0$ we have $y''+x^2y+e^xy=y''(x)+0y'(x)+(x^2+e^x)y(x)$ and $a_1(x)=0\\ a_2(x)=x^2+e^x$ Hence, the given problem has a unique solution. Obtain $y(0)=0$ for all x We have $y''(x)+x^2y(x)+e^xy(x)=0+0+0=0$ Hence, the only solution to the initial-value problem is the trivial solution to $y(x)=0$