Solve the differential equations in Ch 14 of Riley, Hobson & Bence

(* F1 on this for a tutorial *)
tutorial/DSolveOverview

(* separable y' = f[x]g[y] *)
DSolve[y'[x]==x+x y[x],y[x],x]

(* exact *)
DSolve[x y'[x]+3x+y[x]==0,y[x],x]

(* exact after introducing integrating factor *)
DSolve[y'[x]==-(2/(y[x]))-((3y[x])/(2x)),y[x],x]

(* linear y' + a[x] y = f[x] *)
DSolve[y'[x]+2x y[x]==4x,y[x],x]

(* dimensional ("homogeneous") with y ~ x: separable after substition of dimensionless variable v = (y/x) *)
DSolve[y'[x]==((y[x])/x)+Tan[((y[x])/x)],y[x],x]

(* dimensional ("isobaric") with (y^2) ~ (x^(-1)): separable after substition of dimensionless variable v = (y^2)x *)
DSolve[y'[x]==-(1/(2x y[x]))(((y[x])^2)+(2/x)),y[x],x]

(* Bernoulli y' + a[x] y = f[x] (y^n) : linear after substitution v = (y^(1-n)) *)
DSolve[y'[x]+((y[x])/x)==2(x^3)((y[x])^4),y[x],x]

(* separable after linear substitution *)
DSolve[y'[x]==(((x+y[x]+1))^2),y[x],x]

(* dimensional after linear substitution *)
Simplify[DSolve[y'[x]==((2x-5y[x]+3)/(2x-4y[x]-6)),y[x],x]]

(* factorizable into 2 separate equations for y' *)
Simplify[DSolve[((x^3)+(x^2)+x+1)((y'[x])^2)-(3(x^2)+2x+1)y[x]y'[x]+2x ((y[x])^2)==0,y[x],x]]

(* x = f(y,y') *)
(* mathematica does not find all solutions *)
DSolve[6((y[x])^2)((y'[x])^2)+3x y'[x]-y[x]==0,y[x],x]

(* here's the solution that mathematica does not find *)
Simplify[(6((y[x])^2)((y'[x])^2)+3x y'[x]-y[x])/.{y[x]->-((3^(1/3))/2)(x^(2/3)),y'[x]->(∂_x)(-((3^(1/3))/2)(x^(2/3)))}]

(* y = f(x,y') : equivalent to x = f(y,y') with x ↔ y *)
(* mathematica does not find all solutions *)
DSolve[x ((y'[x])^2)+2x y'[x]-y[x]==0,y[x],x]

(* here's the solution that mathematica does not find *)
Simplify[(x ((y'[x])^2)+2x y'[x]-y[x])/.{y[x]->-x,y'[x]->(∂_x)(-x)}]

(* Clairaut equation y = x y' + f(y') is a particular case of y = f(x,y') *)
(* mathematica does not find all solutions *)
DSolve[((y'[x])^2)+x y'[x]-y[x]==0,y[x],x]

(* here's the solution that mathematica does not find *)
Simplify[(((y'[x])^2)+x y'[x]-y[x])/.{y[x]->-(1/4)(x^2),y'[x]->(∂_x)(-(1/4)(x^2))}]