If the solution of differential equation x2d2 y-d x2-2 x d y-d x-12 y is y-A xm-B x-n then the value of m-n- if m- n - N is

Y=Ax^m+Bx^ n ⇒dy/dx=Amx^m 1 nBx^ n 1 ⇒d^2y/dx^2=Am(m 1)x^m 2+n(n+1)Bx^ n 2 Putting these values in x^2d^2y/dx^2+2xdy/dx=12y We get, Am(m 1)x^m+n(n+1)Bx^ n+ ...

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Byju's Answer

Standard XII

Mathematics

Particular Solution of a Differential Equation

If the soluti...

Question

If the solution of differential equation $x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} = 12 y$ is $y = A x^{m} + B x^{- n}$ then the value of $m + n$ , if $m, n \in N$ is

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Solution

$y = A x^{m} + B x^{- n}$

$\Rightarrow \frac{d y}{d x} = A m x^{m - 1} - n B x^{- n - 1}$

$\Rightarrow \frac{d^{2} y}{d x^{2}} = A m (m - 1) x^{m - 2} + n (n + 1) B x^{- n - 2}$

Putting these values in $x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} = 12 y$

We get, $A m (m - 1) x^{m} + n (n + 1) B x^{- n} + 2 A m x^{m} - 2 n B x^{- n} = 12 (A x^{m} + B x^{- n})$

$\Rightarrow m (m + 1) A x^{m} + n (n - 1) B x^{- n} = 12 (A x^{m} + B x^{- n})$

$\Rightarrow m (m + 1) = 12$ or $n (n - 1) = 12$

$\Rightarrow m = 3, - 4$ or $n = 4, - 3$

$\Rightarrow m + n = 3 + 4 = 7$

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If the solution of differential equation x2d2ydx2+2xdydx=12y is y=Axm+Bx−n then the value of m+n, if m,n∈N is

If the solution of differential equation $x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} = 12 y$ is $y = A x^{m} + B x^{- n}$ then the value of $m + n$ , if $m, n \in N$ is