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Evaluate `int_0^oo5^(-4x^2)dx`

Concept: Exact Differential Equations

Evaluate `(d^4y)/(dx^4)+2(d^2y)/(dx^2)+y=0`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

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Evaluate `int_0^1sqrt(sqrtx-x)dx`

Concept: Differentiation Under Integral Sign with Constant Limits of Integration

Solve : `(1+log x.y)dx +(1+x/y)`dy=0

Concept: Exact Differential Equations

Evaluate I = `int_0^1 int_0^(sqrt(1+x^2)) (dx.dy)/(1+x^2+y^2)`

Concept: Double Integration‐Definition

Find the volume of the paraboloid `x^2+y^2=4z` cut off by the plane 𝒛=𝟒

Concept: Triple Integration Definition and Evaluation

If 𝒚 satisfies the equation `(dy)/(dx)=x^2y-1` with `x_0=0, y_0=1` using Taylor’s Series Method find 𝒚 𝒂𝒕 𝒙= 𝟎.𝟏 (take h=0.1).

Concept: Taylor’S Series Method

Evaluate `int int int sqrt(1-x^2/a^2-y^2/b^2-x^2/c^2 )`dx dy dz over the ellipsoid `x^2/a^2+y^2/b^2+z^2/c^2=1.`

Concept: Triple Integration Definition and Evaluation

Evaluate `(2x+1)^2(d^2y)/(dx^2)-2(2x+1)(dy)/(dx)-12y=6x`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

A resistance of 100 ohms and inductance of 0.5 henries are connected in series With a battery of 20 volts. Find the current at any instant if the relation between L,R,E is L `(di)/(dt)+Ri=E.`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `(D^3+1)^2y=0`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve the ODE `(y+1/3y^3+1/2x^2)dx+(x+xy^2)dy=0`

Concept: Exact Differential Equations

Use Taylor’s series method to find a solution of `(dy)/(dx) =1+y^2, y(0)=0` At x = 0.1 taking h=0.1 correct upto 3 decimal places.

Concept: Taylor’S Series Method

Solve `(D^3+D^2+D+1)y=sin^2x`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve the ODE `(D-1)^2 (D^2+1)^2y=0`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Evaluate `int_0^1 int_0^(x2) y/(ex) dy dx`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Find the volume enclosed by the cylinder `y^2=x` and `y=x^2` Cut off by the planes z = 0, x+y+z=2.

Concept: Triple Integration Definition and Evaluation

Evaluate `int_0^1( x^a-1)/log x dx`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `(1+x)^2(d^2y)/(dx^2)+(1+x)(dy)/(dx)+y=4cos(log(1+x))`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Find the length of cycloid from one cusp to the next , where `x=a(θ + sinθ) , y=a(1-cosθ)`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function