Determine order and degree (if defined) of differential equation \(\frac{d^2y}{dx^2}=\cos 3x+\sin3x\)
\(\frac{d^2y}{dx^2}=\cos 3x+\sin3x\)
\(\Rightarrow\frac{d^2y}{dx^2}-\cos3x-\sin3x=0\)
The highest order derivative present in the differential equation is \(\frac{d^2y}{dx^2}.\)
Therefore, its order is two.
It is a polynomial equation in \(\frac{d^2y}{dx^2}\) and the power raised to \(\frac{d^2y}{dx^2}\) is 1.
Hence, its degree is one.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
The equation that helps us to identify the type and complexity of the differential equation is the order and degree of a differential equation.
The highest order of the derivative that appears in the differential equation is the order of a differential equation.
The highest power of the highest order derivative that appears in a differential equation is the degree of a differential equation. Its degree is always a positive integer.
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