Solve: y'' + y = 0, y(0) = 5, y(pi/4) = 3. | Mathematics I 2077

Mathematics I20775 marksNumerical: Second order ODESecond order linear differential equations

Solve: $y'' + y = 0$ , $y(0) = 5$ , $y(\pi/4) = 3$ . [5]

Solve: $y'' + y = 0$, $y(0) = 5$, $y(\pi/4) = 3$. [5]

ODE: $y'' + y = 0$ - Condition 1: $y(0) = 5$ - Condition 2: $y(\pi/4) = 3$ $m^2 + 1 = 0 \implies m = \pm i$ Complex roots with $\alpha = 0$ , $\beta = 1$ . $y = e^{\alpha x}(A\cos\beta x + B\sin\beta x) = A\cos x + B\sin x$ $y(0) = A\cos 0 + B\sin 0 = A = 5$ $\therefore A = 5$ $$y!\left(...

ODE: $y'' + y = 0$ - Condition 1: $y(0) = 5$ - Condition 2: $y(\pi/4) = 3$ $$m^2 + 1 = 0 \implies m = \pm i$$ Complex roots with $\alpha = 0$, $\beta = 1$. $$y = e^{\alpha x}(A\cos\beta x + B\sin\beta x) = A\cos x + B\sin x$$ $$y(0) = A\cos 0 + B\sin 0 = A = 5$$ $$\therefore A = 5$$ $$y!\left(...