/sci/ - Science & Math

>>11466210
>in order to solve a symbolic system of equations (we're doing Laplace transforms) you need to use Kramer's rule.
Nah, you can use row operations just like any other system. He probably just didn't know hand-held calculators had that feature. The result is valid.

>>>/wsr/784888
Start by letting [math] \mathbf{F}(x)=\mathbf{U}'(x) [/math]. Then with the fundamental theorem of calculus and a very small amount of rearranging, you get [eqn] \mathbf{U}'(x)-\frac{1}{x}\mathbf{U}(x)=xe^x\mathbf{A}=\mathbf{U}'(x)+f(x)\mathbf{U}(x)=\mathbf{g}(x) [/eqn] which is just a typical linear ODE. Use an integrating factor to solve this. Specifically, if [math] \mu(x)=\exp\int f(x)\text{ d}x [/math] then [math] \mathbf{U}=[\int \mu \mathbf{g} \text{ d}x+\mathbf{C}]/\mu[/math] (you can find a derivation of this in any ODE textbook). So we have [math] \mu=\exp\int\frac{\text{d} x}{-x}=-x [/math] then [eqn] \mathbf{U}(x)=\frac{\int-x\cdot xe^x\mathbf{A}\text{ d}x+\mathbf{C}}{-x}=\frac{\mathbf{A}e^x(x^2-2x+2)+\mathbf{K}}{x} [/eqn] where K is the arbitrary constant vector we picked up thru integration. Now differentiate U to get F.

>>11445037
You must be new here.