21. With K as a constant, the solution possible for the first order differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {{\text{e}}^{ - 3{\text{x}}}}$$ is
22. The solution of the first order differential equation x'(t) = -3x(t), x(0) = x0 is
23. The solution of the equation $${\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{y}} = 0$$ passing through the point (1, 1) is
24. With initial condition x(1) = 0.5, the solution of the differential equation, $${\text{t}}\frac{{{\text{dx}}}}{{{\text{dt}}}} + {\text{x}} = {\text{t}}$$ is
25. For the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 6\frac{{{\text{dx}}}}{{{\text{dt}}}} + 8{\text{x}} = 0$$ with initial conditions x(0) = 1 and $${\left. {\frac{{{\text{dx}}}}{{{\text{dy}}}}} \right|_{{\text{t}} = 0}} = 0,$$ the solution is
26. Transformation to linear form by substituting v = y1 - n of the equation
$$\frac{{{\text{dy}}}}{{{\text{dt}}}} + {\text{p}}\left( {\text{t}} \right){\text{y}} = {\text{q}}\left( {\text{t}} \right){{\text{y}}^{\text{n}}};\,{\text{n}} > 0$$ will be
$$\frac{{{\text{dy}}}}{{{\text{dt}}}} + {\text{p}}\left( {\text{t}} \right){\text{y}} = {\text{q}}\left( {\text{t}} \right){{\text{y}}^{\text{n}}};\,{\text{n}} > 0$$ will be