## 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.

$\text{y''+0.6y'+0.09y=0, y(0)=2.2, y'(0)=0.14, }e^{-0.3x}\text{, }xe^{-0.3x}$

Solution

Verified4.4 (8 ratings)

4.4 (8 ratings)

Step 1

1 of 5We have

$y''+0.6y'+0.09y=0, \; y\left( 0\right)=2.2, \; y'\left( 0\right)=0.14, \; e^{-0.3x}, \; xe^{-0.3x}$

where,

$\dfrac{y_{1}}{y_{2}}=\dfrac{\bcancel{e^{-0.3x}}}{x\bcancel{e^{-0.3x}}}=\dfrac{1}{x} \neq \text{const}$

$\text{and we see that}\; e^{-0.3x} \; \text{and} \; xe^{-0.3x}\; \text{are linearly independent on any interval.}$

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