# The general solution of the ordinary differential equation

#### Subhotosh Khan

##### Super Moderator
Staff member
Since "purported" solutions are given, I would check each of those for given DE.

Please show us what you have tried and exactly where you are stuck.

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#### nasi112

##### Junior Member
First, find the complementary solution for the homogenous equation.

$$\displaystyle y'' + y = 0$$

then you need a particular solution which looks like

$$\displaystyle y_p = y_1 v_1 + y_2 v_2$$

where $$\displaystyle y_1 = \cos x$$ and $$\displaystyle y_2 = \sin x$$

Calculate the Wronskian [ $$\displaystyle W(y_1,y_2)$$ ] of $$\displaystyle y_1$$ and $$\displaystyle y_2$$

then

$$\displaystyle v_1 = -\int \frac{y_2 \sin x \cos x}{W(y_1,y_2)} \ dx$$

And

$$\displaystyle v_2 = \int \frac{y_1 \sin x \cos x}{W(y_1,y_2)} \ dx$$

Finally

$$\displaystyle y = y_c + y_p$$