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 \(C_2\).
- 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
- 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
- State the Principle of Superposition for 2nd order odes. Verify this for the ode: \(y''-5y'-6y=0 \) solution
- Find the Wronskian of the pair of functions: \(t, te^{t} \) solution
- Find the Wronskian of the pair of functions: \(e^t \sin t, e^t \cos t \) solution
- Show that the set \( \{\cosh t, \sinh t\}\) is a fundamental set of solutions for the ode \(y''-y=0 \). solution
- If the Wronskian of two functions \(f\) and \(g\) is \(-4e^{t} \),
and f is given by \(e^{2t}\), find the function g. solution
- If the Wronskian \(W\) of \(f\) and \(g\) is \(t^2e^t \),
and if \(f(t)=t\), find \(g(t)\). solution