2nd order diff Eq

cheeee · March 14th 2008, 10:11 AM

How to solve this second order diff eq, need help!?
2500y''+9y=0;
y(0) = 1
y'(0) = 1
I think you have to substitute with z = dy/dt and reduce it to a separable equation..but somehow the math is still not working out for me.

**Soroban** · March 14th 2008, 12:30 PM

Hello, cheeee!

This is not a "separable" problem . . .

Solve: . $2500y''+9y\:=\:0,\quad<br /> y(0) \,= \,1,\;\;y'(0) \,=\, 1$

The characteristic equation is: . $2500m^2 + 9 \:=\:0$
. . which has roots: . $m \:=\:\pm\frac{3}{50}i$

Hence, the general solution is: . $y \;=\;C_1\cos\left(\frac{3}{50}x\right) + C_2\sin\left(\frac{3}{50}x\right)$

Since $y(0) = 1\!:\;\;C_1\cos(0) + C_2\sin(0) \:=\:1\quad\Rightarrow\quad\boxed{ C_1 \:=\:1}$

Differentiate: . $y' \;=\;-\frac{3}{50}C_1\sin\left(\frac{3}{50}x\right) + \frac{3}{50}C_2\cos\left(\frac{3}{50}x\right)$

Since $y'(0)=1\!:\;\;-\frac{3}{50}C_1\sin(0) + \frac{3}{50}C_2\cos(0) \:=\:1\quad\Rightarrow\quad\boxed{ C_2 = \frac{50}{3}}$

Therefore: . $\boxed{\;y \;=\;\cos\left(\frac{3}{50}x\right) + \frac{50}{3}\sin\left(\frac{3}{50}x\right)\;}$

**TheEmptySet** · March 14th 2008, 12:38 PM

Originally Posted by cheeee

How to solve this second order diff eq, need help!?
2500y''+9y=0;
y(0) = 1
y'(0) = 1
I think you have to substitute with z = dy/dt and reduce it to a separable equation..but somehow the math is still not working out for me.

$50^2y''+3^2y=0$

solve the equation

$50^2m^2+3^2=0$

rewriting we get

$(50m)^2+3^2=(50m)^2-(3i)^2=(50m-3i)(50m+3i)=0$

so the two solutions for m are

$m=\frac{3i}{50}$ or $=\frac{-3i}{50}$

so we end up with the complimentry solution.

$y_c=c_1\cos(\frac{3}{50}t)+c_2\sin{\frac{3}{50}t}$

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