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Exact Solutions >
FirstOrder Partial Differential Equations >
Linear Partial Differential Equations
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1. FirstOrder Linear Partial Differential Equations
1.1. Equations of the Form f(x, y)w_{x} + g(x, y)w_{y} = 0

w_{x} + [f(x)y + g(x)]w_{y} = 0.

w_{x} + [f(x)y + g(x)y^{k}]w_{y} = 0.

w_{x} + [f(x)e^{λy} + g(x)]w_{y} = 0.

f(x)w_{x} + g(y)w_{y} = 0.

[f(y) + amx^{n}y^{m−1}]w_{x}
− [g(x) + anx^{n−1}y^{m}]w_{y} = 0.

[e^{αx}f(y) + cβ]w_{x} − [e^{βy}g(x) + cα]w_{y} = 0.

w_{x} + f(ax + by + c)w_{y} = 0.

w_{x} + f(y/x)w_{y} = 0.

xw_{x} + yf(x^{n}y^{m})w_{y} = 0.

w_{x} + yf(e^{αx}y^{m})w_{y} = 0.

xw_{x} + f(x^{n}e^{αy})w_{y} = 0.
1.2. Equations of the Form f(x, y)w_{x} + g(x, y)w_{y} = h(x, y)

aw_{x} + bw_{y} = f(x).

w_{x} + aw_{y} = f(x)y^{k}.

w_{x} + aw_{y} = f(x)e^{λy}.

aw_{x} + bw_{y} = f(x) + g(y).

w_{x} + aw_{y} = f(x)g(y).

w_{x} + aw_{y} = f(x, y).

w_{x} + [ay + f(x)]w_{y} = g(x).

w_{x} + [ay + f(x)]w_{y} = g(x)h(y).

w_{x} + [f(x)y + g(x)y^{k}]w_{y} = h(x).

w_{x} + [f(x) + g(x)e^{λy}]w_{y} = h(x).

axw_{x} + byw_{y} = f(x, y).

f(x)w_{x} + g(y)w_{y} = h_{1}(x) + h_{2}(y).

f(x)w_{x} + g(y)w_{y} = h(x, y).

f(y)w_{x} + g(x)w_{y} = h(x, y).
1.3. Equations of the Form f(x, y)w_{x} + g(x, y)w_{y} = h(x, y)w + r(x, y)

aw_{x} + bw_{y} = f(x)w.

aw_{x} + bw_{y} = f(x)w + g(x).

aw_{x} + bw_{y} = [f(x) + g(y)]w.

w_{x} + aw_{y} = f(x, y)w.

w_{x} + aw_{y} = f(x, y)w + g(x, y).

axw_{x} + byw_{y} = f(x)w + g(x).

axw_{x} + byw_{y} = f(x, y)w.

xw_{x} + ayw_{y} = f(x, y)w + g(x, y).

f(x)w_{x} + g(y)w_{y} = [h_{1}(x) + h_{2}(y)]w.

f_{1}(x)w_{x} + f_{2}(y)w_{y} = aw + g_{1}(x) + g_{2}(y).

f(x)w_{x} + g(y)w_{y} = h(x, y)w + r(x, y).

f(y)w_{x} + g(x)w_{y} = h(x, y)w + r(x, y).
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