Verify: $y=e^{ax}\sin bx$, and $\dfrac{d^2y}{dx^2}-2a\dfrac{dy}{dx}+(a^2+b^2)y=0$

Verify: y=eaxsin bx, and (d²y)/(dx²)-2a(dy)/(dx)+(a²+b²)y=0

Miscellaneous Exercise 6(II) · Q117

y=e^{ax}\sin bx

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$y=e^{ax}\sin bx$ has characteristic roots $a\pm bi$ , so it solves the standard second-order equation $\dfrac{d^2y}{dx^2}-2a\dfrac{dy}{dx}+(a^2+b^2)y=0$ . Directly: $y'=e^{ax}(a\sin bx+b\cos bx)$ , $y''=e^{ax}\left[(a^2-b^2)\sin bx+2ab\cos bx\right]$ , and substituting confirms e …

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Q.Verify: $y=e^{ax}\sin bx$ , and $\dfrac{d^2y}{dx^2}-2a\dfrac{dy}{dx}+(a^2+b^2)y=0$

Concept understanding — General and Particular Solutions of a Differential Equation

Q.Verify: y=eaxsin⁡bxy=e^{ax}\sin bxy=eaxsinbx, and d2ydx2−2adydx+(a2+b2)y=0\dfrac{d^2y}{dx^2}-2a\dfrac{dy}{dx}+(a^2+b^2)y=0dx2d2y​−2adxdy​+(a2+b2)y=0

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Q.Verify: $y=e^{ax}\sin bx$ , and $\dfrac{d^2y}{dx^2}-2a\dfrac{dy}{dx}+(a^2+b^2)y=0$

Unlock everything free for 14 days