u(x,t)=∑n=1∞Ansin(nπxL)e−k(nπL)2tu open paren x comma t close paren equals sum from n equals 1 to infinity of cap A sub n sine open paren the fraction with numerator n pi x and denominator cap L end-fraction close paren e raised to the exponent negative k open paren the fraction with numerator n pi and denominator cap L end-fraction close paren squared t end-exponent ✅
). The spatial ODE is typically an eigenvalue problem (e.g., Partial Differential Equations with Fourier Ser...
terms on the other. Because they depend on different variables but are equal, both sides must equal a constant, typically denoted as −λnegative lambda This yields two separate ODEs: one for space ( ) and one for time ( both sides must equal a constant
