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Show that by changing the independent variable $x$ to $z$ using the substitution $x^{2}=z$ the relation $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+4\left(x^{4...

Question

Show that by changing the independent variable $x$ to $z$ using the substitution $x^{2}=z$ the relation $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+4\left(x^{4}-\lambda^{2}\right) y=0$ reduces to $z^{2} \frac{d^{2} y}{d z^{2}}+z \frac{d y}{d z}+\left(z^{2}-\lambda^{2}\right) y=0$

Show that by changing the independent variable $x$ to $z$ using the substitution $x^{2}=z$ the relation $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+4\left(x^{4}-\lambda^{2}\right) y=0$ reduces to $z^{2} \frac{d^{2} y}{d z^{2}}+z \frac{d y}{d z}+\left(z^{2}-\lambda^{2}\right) y=0$



Answers

Show that by changing the independent variable $x$ to $z$ using the substitution $x^{2}=z$ the relation $x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+4\left(x^{4}-\lambda^{2}\right) y=0$ reduces to $z^{2} \frac{d^{2} y}{d z^{2}}+z \frac{d y}{d z}+\left(z^{2}-\lambda^{2}\right) y=0$


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