Math  /  Calculus

QuestionFill in the table on the first page with the correct answers to the following 15 questions ( 30 Pts)
1- The result of (Dx+1)(xyex+y)=\left(D_{x}+1\right)\left(x y e^{x+y}\right)= a) ex+y(3xy)e^{x+y}(3 x y) b) ex+y(y+2xy)e^{x+y}(y+2 x y) c) ex+y(y+xy)e^{x+y}(y+x y). d) ex+y(x+2xy)e^{x+y}(x+2 x y)
2-The Euler PDE that has the general solution u(x,y)=F(y+x)+xG(y+x)u(x, y)=F(y+x)+x G(y+x) is a) uxx+2uxy+uyy=0u_{x x}+2 u_{x y}+u_{y y}=0 b) uxx+2uxyuxy=0u_{x x}+2 u_{x y}-u_{x y}=0 c) uxx2uxyuyy=0u_{x x}-2 u_{x y}-u_{y y}=0 d) uxx2uxy+uyy=0u_{x x}-2 u_{x y}+u_{y y}=0
3- The general solution of s+1xq=6xys+\frac{1}{x} q=6 x y is u(x,y)=u(x, y)= a) x2y2+G(y)x+h(x)x^{2} y^{2}+\frac{G(y)}{x}+h(x) b) x2y2+G(y)x+h(y)x^{2} y^{2}+\frac{G(y)}{x}+h(y) c) x2y2+G(x)x+h(y)x^{2} y^{2}+\frac{G(x)}{x}+h(y) d) none
4- The separation of variables method is used to solve a) 1st 1^{\text {st }} degree and 2nd 2^{\text {nd }} order b) 1st 1^{\text {st }} order and 2nd 2^{\text {nd }} degree c) 1st 1^{\text {st }} degree, 2nd 2^{\text {nd }} degree d) 1st 1^{\text {st }} order and 2nd 2^{\text {nd }} order
5- The PDE that has the general solution F(x+z,x+y)=0F(x+z, x+y)=0 is a) q=1pq=1-p b) q=pq=p c) 1+p2=q1+p^{2}=q d) q=1+pq=1+p
6- By using the Charpit's method to solve pq=axyp q=a x y one of the five ratios is dzR\frac{d z}{R}, then the value of R=R= a) 2axy2 a x y b) xy-x y. c) xyx y d) 2axy-2 a x y
7- The value of (Dx2+Dy2)e2xy\left(D_{x}^{2}+D_{y}^{2}\right) e^{2 x-y} is a) 3e2xy3 e^{2 x-y} b) 4e2xy4 e^{2 x-y} c) 5e2xy5 e^{2 x-y} d) 2e2xy2 e^{2 x-y}

Studdy Solution
Solve Question 7:
Calculate the value of (Dx2+Dy2)e2xy (D_x^2 + D_y^2) e^{2x-y} .
First, find the second derivative with respect to x x :
Dx2(e2xy)=4e2xy D_x^2(e^{2x-y}) = 4e^{2x-y}
Then, find the second derivative with respect to y y :
Dy2(e2xy)=e2xy D_y^2(e^{2x-y}) = e^{2x-y}
Add the results:
(Dx2+Dy2)e2xy=4e2xy+e2xy=5e2xy (D_x^2 + D_y^2)e^{2x-y} = 4e^{2x-y} + e^{2x-y} = 5e^{2x-y}
The correct answer is (c) 5e2xy 5 e^{2x-y} .
The answers to the questions are:
1. b
2. d
3. a
4. d
5. a
6. d
7. c

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