differenzialgleichungen-bsp

3.10) a) $y^{\prime }-3x=1$ $y^{\prime }=3x+1$ $y(x)=\int (3x+1)dx$ $y(x)=\frac{3}{2}{x}^2+x+C$ c) $y^{\prime }-4x+x^2=1$ $y^{\prime }=-x^2+4x+1$ $y=-\frac{x^3}{3}+2x^2+x+C$ e) $y^{\prime \prime }+6x-3=0$ $y^{\prime \prime }=-6x+3$ $y^{\prime }=3x^2+3x+C_1$ $y=-x^3+\frac{3x}{2}+C_{1}x+C_2$ 3.11) b) $y=x+x^2+C$ $y^{\prime }=-2x+1$ 3.12) b) $y^{\prime }+x^2=x+1;y(6)=0$ c) $y^{\prime \prime }-x+1=0;y(1)=0,y^{\prime }(1)=0$ umstellen $y^{\prime \prime }=x-1$ integrieren $y^{\prime }\frac{x^2}{2}-x+C_1$ nochamal $y=\frac{x^3}{6}-\frac{x^2}{2}+C_1\ast x+C_2$ $y^{\prime }(1)=0:0=\frac{1^2}{2}-1+C_1\to C_1=\frac{1}{2}$ $c{C}_2=-\frac{1}{6}$ spez lösung: $y=-\frac{x^3}{6}+\frac{x^2}{2}+\frac{x}{2}-\frac{1}{6}$

3.13) $f(1)=0$ $y^{\prime }\frac{1}{x}$ $y=\ln |x|+C$ $0=\ln |1|+C$ $y=\ln |x|$ 3.15) mittlere Beschl: $a(t)=0.5\frac{m}{s^2}$ $a(t)=v^{\prime }(t)=s^{\prime \prime }(t)$ a, s(t)=? $a(t)=v^{\prime }(t)=0,5$ $v(t)=0.5t+C_1$ algemeine lösung: $s(t)=\frac{t^2}{4}+C_{1}t+C_2$ b) $v(0)=0$ $s(0)=0$ $0=\frac{0}{4}+C_1\ast 0+C_2\to C_2=0$ $0=0,5\ast 0+C_1\to C_1=0$ $s(t)=\frac{t^2}{4}=0.25t^2$ c) bis 200km/h = 55,5m/s $v(t)=55.5=0.5(t)\to t=111.1s$ 3.38 c) $\frac{1}{2}\ast p_0=p_0\ast e^{-\frac{h}{7.99}}$