Verify that $y=x^m$ is a solution of $x^2\dfrac{d^2y}{dx^2}-mx\dfrac{dy}{dx}+my=0$.

Verify that y=xm is a solution of x²(d²y)/(dx²)-mx(dy)/(dx)+my=0.

Exercise 6.3 · Q30

y=x^m

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Differentiate $y=x^m$ twice: $\dfrac{dy}{dx}=mx^{m-1}$ , $\dfrac{d^2y}{dx^2}=m(m-1)x^{m-2}$ . Substitute into $x^2\dfrac{d^2y}{dx^2}-mx\dfrac{dy}{dx}+my$ : $x^2\cdot m(m-1)x^{m-2}-mx\cdot mx^{m-1}+m\cdot x^m=m(m-1)x^m-m^2x^m+mx^m=x^m\big[m^2-m-m^2+m\big]=0$ . So th …

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Q.Verify that $y=x^m$ is a solution of $x^2\dfrac{d^2y}{dx^2}-mx\dfrac{dy}{dx}+my=0$ .

Concept understanding — General and Particular Solutions of a Differential Equation

Q.Verify that y=xmy=x^my=xm is a solution of x2d2ydx2−mxdydx+my=0x^2\dfrac{d^2y}{dx^2}-mx\dfrac{dy}{dx}+my=0x2dx2d2y​−mxdxdy​+my=0.

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Q.Verify that $y=x^m$ is a solution of $x^2\dfrac{d^2y}{dx^2}-mx\dfrac{dy}{dx}+my=0$ .

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