## Related questions with answers

Question

(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

$4x^2y''-3y=0, y(1)=-3, y(1)=0, x^{3/2},x^{-1/2}$

Solution

Verified4 (5 ratings)

4 (5 ratings)

Step 1

1 of 5We have

$4x^{2}y''-3y=0, \; y\left(1\right)=3, \; y'\left(1\right)=0, \; x^{\frac{3}{2}},\; x^{-\frac{1}{2}}$

where,

$\dfrac{y_{1}}{y_{2}}=\dfrac{x^{\frac{3}{2}}}{x^{-\frac{1}{2}}}=x^{2} \neq \text{const}$

$\text{and we see that} \; x^{\frac{3}{2}} \; \text{and} \; x^{-\frac{1}{2}}$ are linearly independent on any interval.

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