equation1=((∂u(x,y))/(∂x))+x ((∂u(x,y))/(∂y))==sin(x);

sol=u/.DSolve[equation1, u, {x,y}][[1]]/.(Subscript[\[ConstantC], 1])[t_]->t

equation1 /.{u-> sol}

DSolve[D[u[x,y],x]==1, u[x,y],{x,y}]

z:=u[x,y]

p:=D[u[x,y],x]

q:=D[u[x,y],y]

eqn=2*p+3*q +z==0;

sol=DSolve[eqn, u, {x,y}]

eqn/.sol[[1]]//Simplify

particularsolution = u[x,y]/.sol[[1]]/.(C[1])[a_]-> Sin[a]

Plot3D[particularsolution, {x,-2,2}, {y,-7,7}, PlotPoints -> 30]

DSolve[D[u[x,y],x]+D[u[x,y],y]==0,u[x,y],{x,y}]

eqn=7*p+3*q +z==x+y;

sol=u[x,y]/.DSolve[eqn, u[x,y], {x,y}][[1]]//Expand

eqn=Sin[x]*p+E^x*q==0;

sol=DSolve[eqn, u, {x,y}]

eqn/.sol[[1]]//Simplify

eqn=p+x*q==Cos[x];

sol=DSolve[eqn, u, {x,y}]

eqn=p+x*q==z^2+5;

sol=DSolve[eqn, u, {x,y}]

eqn/.sol[[1]]//Simplify

BurgersEquation =p+z q==0;

sol=DSolve[BurgersEquation, u, {x,y}]

p1=p/.Solve[D[sol[[1]],x], p][[1]];

q1=q/.Solve[D[sol[[1]],y], q][[1]];

p1+z*q1

z:=u[x,y]

p:=D[u[x,y],x]

q:=D[u[x,y],y]

eqn =p*q==1;

sol=DSolve[eqn, u, {x,y}]

eqn/.sol

oneparametersol = u[x,y]== (u[x,y]/.sol[[1]]/.(C[2])-> 5*(C[1]))

oneparameterenvelope = Eliminate[{oneparametersol, D[oneparametersol,(C[1])]}, {(C[2])}]

p1=D[u[x,y],x]/.Solve[D[oneparameterenvelope, x], D[u[x,y],x]][[1]]

q1=D[u[x,y],y]/.Solve[D[oneparameterenvelope, y], D[u[x,y],y]][[1]]

FullSimplify[p1*q1, {oneparameterenvelope}]

sol=DSolve[4z+p^2+q^2==4, u, {x,y}]

twoparameterfamily = u[x,y]== (u[x,y]/.sol[[1]])

envelopeoftwoparameterfamily = Eliminate[{twoparameterfamily, D[twoparameterfamily, (C[1])], D[twoparameterfamily,(C[2])]}, {(C[1]),(C[2])}]

Eikonal=D[u[x,y],x]^2+D[u[x,y],y]^2==1;

sol=DSolve[Eikonal, u, {x,y}]

Eikonal/.sol

Clairaut =z== x*p+y*q+ 2p*q*Sqrt[1-p^2];

sol=DSolve[Clairaut, u, {x,y}]

Clairaut/.sol

Separable = p^2+a*q==x+3y;

sol=DSolve[Separable, u, {x,y}]

Separable/.sol//Simplify

MissingIndependentVariables = a*p^2+b*p*q==c*z^2;

sol = DSolve[MissingIndependentVariables, u, {x,y}]

MissingIndependentVariables/.sol//Simplify

DSolve[x*y*p*q==1, u[x,y],{x,y}]

DSolve[(y*p-x*q)^2+a*(x*p+y*q)==b, z, {x,y}]

LegendreTransformable = y*p*q-z*p+a*q==0;

sol = DSolve[LegendreTransformable, u, {x,y}]

LegendreTransformable /.sol//Simplify

sol=DSolve[(1+q^2)*z==p*x,u, {x,y}]

p=D[u[x,y],x]/.Solve[D[sol[[1]],x], D[u[x,y],x]][[1]]

q=D[u[x,y],y]/.Solve[D[sol[[1]],y], D[u[x,y],y]][[1]]

(1+q^2)*z-p*x//Simplify

LaplaceEquation=D[u[x,y], {x,2}]+D[u[x,y], {y, 2}]==0;

DSolve[LaplaceEquation,u[x,y],{x,y}]

a= 3;b=1;c=5;b^2-4a*c

eqn = a*D[u[x,y],{x,2}]+b*D[u[x,y], x,y]+c*D[u[x,y], {y,2}]==0;

sol = DSolve[eqn, u, {x,y}]

eqn/.sol//Simplify

WaveEquation=D[u[x,t],{x,2}]-D[u[x,t],{t,2}]==0;

DSolve[WaveEquation , u[x,t], {t,x}]

a= 2;b=7;c=-1;b^2-4a*c

eqn = a*D[u[x,y],{x,2}]+b*D[u[x,y], x,y]+c*D[u[x,y], {y,2}]==0;

sol = DSolve[eqn, u, {x,y}]

eqn/.sol//Simplify

a= 3;b=30;c= 75;b^2-4a*c

eqn = a*D[u[x,y],{x,2}]+b*D[u[x,y], x,y]+c*D[u[x,y], {y,2}]==0;

sol = DSolve[eqn, u, {x,y}]

eqn/.sol//Simplify