The general solution of the ordinary differential equation

First, find the complementary solution for the homogenous equation.

[MATH]y'' + y = 0[/MATH]
then you need a particular solution which looks like

[MATH]y_p = y_1 v_1 + y_2 v_2[/MATH]
where [MATH]y_1 = \cos x[/MATH] and [MATH]y_2 = \sin x[/MATH]
Calculate the Wronskian [ [MATH]W(y_1,y_2)[/MATH] ] of [MATH]y_1[/MATH] and [MATH]y_2[/MATH]
then

[MATH]v_1 = -\int \frac{y_2 \sin x \cos x}{W(y_1,y_2)} \ dx[/MATH]
And

[MATH]v_2 = \int \frac{y_1 \sin x \cos x}{W(y_1,y_2)} \ dx[/MATH]
Finally

[MATH]y = y_c + y_p[/MATH]
 
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