If y-ea x-cos b x- then prove that d2 y-d x2-2 a d y-d x-a2-b2 y-0

Y = e^ax. cos bx dy/dx=ae^ax.cos bx be^ax.sin bx...........(i) dy/dx = ay be^ax.sin bx d^2 y/dx^2= a dy/dx b(ae^ax.sin bx + be^ax.cos bx) d^2 y/dx^2= a dy/dx ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Integration by Parts

If y=ea x·cos...

Question

If $y = e^{a x}$ . cos bx, then prove that $\frac{d^{2} y}{d x^{2}} - 2 a \frac{d y}{d x} + (a^{2} + b^{2}) y = 0$

Open in App

Solution

$y = e^{a x} . c o s b x$

$\frac{d y}{d x} = a e^{a x} . c o s b x - b e^{a x} . s i n b x . . . . . . . . . . . (i) \frac{d y}{d x} = a y - b e^{a x} . s i n b x \frac{d^{2} y}{d x^{2}} = a \frac{d y}{d x} - b (a e^{a x} . s i n b x + b e^{a x} . c o s b x) \frac{d^{2} y}{d x^{2}} = a \frac{d y}{d x} - b a e^{a x} . s i n b x - b^{2} e^{a x} . c o s b x \frac{d^{2} y}{d x^{2}} = a \frac{d y}{d x} - a (a y - \frac{d y}{d x}) - b^{2} y$

[Substituting $b e^{a x}$ sin bx from (i)]

$\frac{d^{2} y}{d x^{2}} = a \frac{d y}{d x} - a^{2} y + a \frac{d y}{d x} - b^{2} y ∴ \frac{d^{2} y}{d x^{2}} - 2 a \frac{d y}{d x} + (a^{2} + b^{2}) y = 0$

Hence Proved

Suggest Corrections

Similar questions

Q. If

y = e^{a x} cos b x

, then prove that

\frac{d^{2} y}{d x^{2}} - 2 a \frac{d y}{d x} + (a^{2} + b^{2}) y = 0

Q. If

y = e^{a x} cos b x

, then find the value of

\frac{d^{2} y}{d x^{2}} - 2 a \frac{d y}{d x} + (a^{2} + b^{2}) y

Q. Then

y = e^{a x} s i n b x

solve the equation:

\frac{d^{2} y}{d x^{2}} - 2 a

\frac{d y}{d x} + (a^{2} + b^{2}) y = 0

If $y = P e^{a x} + Q e^{b x}$ , then prove that $\frac{d^{2} y}{d x^{2}} - (a + b) \frac{d y}{d x} + a b y = 0$

Q. If

y = sin (sin x)

, prove that

\frac{d^{2} y}{d x^{2}} + tan x \frac{d y}{d x} + y {cos}^{2} x = 0

Join BYJU'S Learning Program

If y=eax. cos bx, then prove that d2ydx2−2adydx+(a2+b2)y=0

If $y = e^{a x}$ . cos bx, then prove that $\frac{d^{2} y}{d x^{2}} - 2 a \frac{d y}{d x} + (a^{2} + b^{2}) y = 0$