Differential Equation
87634
\(u \equiv u(x, y)=\sin (y+a x)-(y+a x)^{2}\), then it implies
1 \(u_{x x}=a^{2} \cdot u_{y y}\)
2 \(u_{y y}=a^{2} u_{z x}\)
3 \(u_{x x}=-a^{2} u_{y y}\)
4 \(u_{y y}=-a^{2} u_{x x}\)
Explanation:
(A) : Given, that-
\(u=\sin (y+a x)-(y+a x)^{2} \tag{i}\)
On differentiating partially w.r.t.x,
\(u_{x}=\cos (y+a x) a-2(y+a x) a\)
Again differentiating partially. \(w_{2}\) r.t.x
\(\mathrm{u}_{\mathrm{xx}}=-\sin (\mathrm{y}+\mathrm{ax}) \mathrm{a}^{2}-2 \mathrm{a}^{2} \tag{ii}\)
On differentiating partially equation (i) w.r.t.y,
\(u_{y}=\cos (y+a x)-2(y+a x)\)
\(u_{y y}=-\sin (y+a x)-2\)
\(u_{y y}=-\sin (y+a x)-2 \tag{iii}\)
\(\therefore\) From equation (ii) and (iii), we get \(\mathrm{u}_{\mathrm{xx}}=\mathrm{a}^{2} \mathrm{u}_{\mathrm{yy}}\)