07 Known Soln
Consider a homogeneous 2nd order DE.
\[ y'' + P y' + Q y = 0 \]
Let \(y_1(x)\) be the known solution of it.
To find another linear-independent solution \(y_2(x)\)
- Let
\[ \begin{aligned} v &= \int \frac{1}{ {(y_1)}^2 \times e^{ \int P dx} } \\ y_2 &= v \cdot y_1 \end{aligned} \]
- Now, the general solution \(y(x) = c_1 y_1(x) + c_2 y_2(x)\)
Special Cases¶
(not important)
Legendre DE¶
\[ (1-x^2)y'' - 2xy' + k(k+1) y = 0 \]
where \(k\) = const
Bessel’s Equation¶
\[ x^2 y'' + xy' + (x^2 - k^2) y = 0 \]
\(k\) = const