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Let <math>f</math> and <math>g</math> be the two functions defined in | Let <math>f</math> and <math>g</math> be the two functions defined in | ||
Problem [[exercise:7bc859a7a4 |Exercise]] (see also Problem [[exercise:0cba72917a |Exercise]]). | Problem [[exercise:7bc859a7a4 |Exercise]] (see also Problem [[exercise:0cba72917a |Exercise]]). | ||
<ul style{{=}}"list-style-type:lower-alpha" | <ul style{{=}}"list-style-type:lower-alpha"> | ||
<li> | <li> | ||
Evaluate <math>f(0)</math>, <math>f^\prime(0)</math>, <math>g(0)</math>, | Evaluate <math>f(0)</math>, <math>f^\prime(0)</math>, <math>g(0)</math>, | ||
and <math>g^\prime(0)</math>.</li> | and <math>g^\prime(0)</math>.</li> | ||
<li> | <li> | ||
Show that <math>f</math> and <math>g</math> are both solutions of the | Show that <math>f</math> and <math>g</math> are both solutions of the | ||
differential equation <math>\frac{d^2y}{dx^2} + y = 0</math>.</li> | differential equation <math>\frac{d^2y}{dx^2} + y = 0</math>.</li> | ||
<li>Write the general solution of the differential equation | <li>Write the general solution of the differential equation | ||
in | in (b), and thence, using the results of | ||
part | part (a), show that <math>f(x) = \sin x</math> | ||
and that <math>g(x) = \cos x</math>.</li> | and that <math>g(x) = \cos x</math>.</li> | ||
</ul> | </ul> | ||
Latest revision as of 03:18, 25 November 2024
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[/math]
Let [math]f[/math] and [math]g[/math] be the two functions defined in Problem Exercise (see also Problem Exercise).
- Evaluate [math]f(0)[/math], [math]f^\prime(0)[/math], [math]g(0)[/math], and [math]g^\prime(0)[/math].
- Show that [math]f[/math] and [math]g[/math] are both solutions of the differential equation [math]\frac{d^2y}{dx^2} + y = 0[/math].
- Write the general solution of the differential equation in (b), and thence, using the results of part (a), show that [math]f(x) = \sin x[/math] and that [math]g(x) = \cos x[/math].