Basic Concepts 


 Principle of Superpositon: If \(y_1\) and \(y_2\) are two solutions of a linear, homogeneous differential equation then so is \(C_1y_1+C_2y_2\) for arbitrary constants \(C_1\) and \(C_2\). 

Finding Wronskian of two or three solutions (video)

  1. Show that \(e^t\) and \(e^{-t}\) are solutions of the ode \(y''-y=0\). Moreover show that \(C_1e^t+C_2e^{-t}\) is also a solution of the ode for any constants \(C_1\) and \(C_2\).  solution

  2. Show that \(y_1=e^x\cos x\) and \(y_2=e^{x}\sin x\) are solutions of the differential equation \(y''-2y'+2y=0\). Show that \(C_1e^x\cos x + C_2e^{x} \sin x\), where \(C_1\) and \(C_2\) are arbitrary constants, is also a solution of the diff eqn.  solution

  3. State the Principle of Superposition for 2nd order odes. Verify this for the ode: \(y''-5y'-6y=0 \)  solution

  4. Find the Wronskian of the pair of functions: \(t, te^{t} \)    solution

  5. Find the Wronskian of the pair of functions:  \(e^t \sin t,  e^t \cos t \)    solution

  6. Show that the set \( \{\cosh t, \sinh t\}\) is a fundamental set of solutions for the ode \(y''-y=0 \).   solution

  7. If the Wronskian of two functions \(f\) and \(g\) is \(-4e^{t} \), and f is given by \(e^{2t}\), find the function g.   solution

  8. If the Wronskian \(W\) of \(f\) and \(g\) is \(t^2e^t \), and  if \(f(t)=t\), find \(g(t)\).   solution