Question

Найти решения задачи Коши и

Answer 1

Дифференциальное уравнение второго порядка.
$y''+ \pi^2*y=\frac{\pi^2}NO NUMERIC NOISE KEY 1112\\\lambda^2+\pi^2=0\\\lambda^2=-\pi^2\\\lambda_NO NUMERIC NOISE KEY 1111=^+_-\pi i\\y=C_1cos\pi x+c_2sin\pi x\\c_1=C_1(x);C_2=C_2(x)\\\beginNO NUMERIC NOISE KEY 1110c_1'(x)cos\pi x+c_2'(x)sin\pi x=0\\-C_1'(x)sin\pi x+c_2'(x)cos\pi> <img src=$
$y=3cos\pi x+cos\pi x*ln|cos\pi x|+\pi x*sin\pi x$
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Дифференциальное уравнение второго порядка.
$y''+ \pi^2*y=\frac{\pi^2}{sin\pi x}\\\lambda^2+\pi^2=0\\\lambda^2=-\pi^2\\\lambda_{1,2}=^+_-\pi i\\Y=C_1cos\pi x+C_2sin\pi x\\C_1=C_1(x);C_2=C_2(x)\\\begin{cases}C_1'(x)cos\pi x+C_2'(x)sin\pi x=0\\-C_1'(x)sin\pi x+C_2'(x)cos\pi x=\frac{\pi}{sin\pi x}\end{cases}\\W= \left[\begin{array}{cc}cos\pi x&sin\pi x\\-sin\pi x&cos\pi x\end{array}\right] =1\\W_1=\left[\begin{array}{cc}0&sin\pi x\\\frac{\pi}{sin\pi x}&cos\pi x\end{array}\right]=-\pi$
$W_2=\left[\begin{array}{cc}cos\pi x&0\\-sin\pi x&\frac{\pi}{sin\pi x}\end{array}\right]=\pi ctg\pi x\\C_1'(x)=\frac{W_1}{W}=-\pi\\C_2'(x)=\frac{W_2}{W}=\pi ctg\pi x\\C_1(x)=-\pi\int dx=-\pi x+C_1\\C_2(x)=\pi\int ctg\pi xdx=ln|sin\pi x|+C_2\\y=C_1cos\pi x+C_2sin\pi x-\pi x*cos\pi x+sin\pi x*ln|sin\pi x|\\y(\frac{1}{2})=1\\\frac{1}{2}=C_2\\y'=-\pi C_1sin\pi x+\pi C_2cos\pi x+\pi^2 xsin\pi x+\pi cos\pi x*ln|sin\pi x|\\y'(\frac{1}{2})=\frac{\pi^2}{2}\\0=C_1$
$y=\frac{1}{2}sin\pi x-\pi x*cos\pi x+sin\pi x*ln|sin\pi x|$