Welche der folgenden Differentialgleichungen 1 . Ordnung sind linear, welche nichtlinear? Unterscheiden Sie dabei die linearen Differentialgleichungen nach homogenen und inhomogenen Differentialgleichungen. a) \(y^{\prime}=x y\) b) \(\quad x^{3} y^{\prime}-y=2 x y^{2}\) c) \(y^{\prime}-2 y=\sin x\) d) \(y^{\prime} \cdot \cos x-y+\sin x=1\) e) \(\quad y^{\prime} y^{2}+x^{2}=1\) f) \(y^{\prime}=\sqrt{y}\) g) \(L \frac{d i}{d t}+R i=u(t)\) h) \(y^{\prime}-x\left(1+y^{2}\right)\) i) \(x y^{\prime}+y=\ln x\) j) \(m \dot{v}+k v=m g\) k) \(\quad y^{\prime} \sqrt{y}-x=0\) l) \(y^{\prime}=5 x^{4}(y+1)\)

List the given differential equations: a) \(y' = xy\) b) \(x^3 y' - y = 2xy^2\) c) \(y' - 2y = \sin x\) d) \(y' \cos x - y + \sin x = 1\) e) \(y' y^2 + x^2 = 1\) f) \(y' = \sqrt{y}\) g) \(L \frac{di}{dt} + Ri = u(t)\) h) \(y' - x(1 + y^2)\) i) \(x y' + y = \ln x\) j) \(m \dot{v} + kv = mg\) k) \(y' \sqrt{y} - x = 0\) l) \(y' = 5x^4(y + 1)\)

A first-order differential equation is linear if it can be written as \(a_1(x) y' + a_0(x) y = b(x)\). If any term involving nonlinear functions of \(y\) or \(y'\) appears, it is nonlinear.Linear: no products, powers, or other nonlinear functions of \(y\) or \(y'\).

a) \(y' = xy\) is nonlinear (product of \(y\) and \(y'\)).b) \(x^3 y' - y = 2xy^2\) is nonlinear (\(y^2\) term).c) \(y' - 2y = \sin x\) is linear (standard form, non-homogeneous).d) \(y' \cos x - y + \sin x = 1\) is linear (standard form, non-homogeneous).e) \(y' y^2 + x^2 = 1\) is nonlinear (\(y^2\) term).f) \(y' = \sqrt{y}\) is nonlinear (\(\sqrt{y}\) term).g) \(L \frac{di}{dt} + R i = u(t)\) is linear (standard form, non-homogeneous).h) \(y' - x(1 + y^2) = 0\) is nonlinear (\(y^2\) term).i) \(x y' + y = \ln x\) is linear (standard form, non-homogeneous).j) \(m \dot{v} + kv = mg\) is linear (standard form, non-homogeneous).k) \(y' \sqrt{y} - x = 0\) is nonlinear (\(\sqrt{y}\) term).l) \(y' = 5 x^4 (y + 1)\) is nonlinear (product of \(y\) and \(y'\)).

For linear equations, check homogeneity:c) \(y' - 2y = \sin x\) is inhomogeneous (\(\sin x\) is nonzero).d) \(y' \cos x - y + \sin x = 1\) is inhomogeneous (1 is nonzero).g) \(L \frac{di}{dt} + Ri = u(t)\) is inhomogeneous (\(u(t)\) is nonzero).i) \(x y' + y = \ln x\) is inhomogeneous (\(\ln x\) is nonzero).j) \(m \dot{v} + kv = mg\) is inhomogeneous (\(mg\) is nonzero).