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

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Solve `(D^2+2)y=e^xcosx+x^2e^(3x)`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

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

Concept: Exact Differential Equations

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

Concept: Exact Differential Equations

Solve `xy(1+xy^2)(dy)/(dx)=1`

Concept: Linear Differential Equations

Solve `(y-xy^2)dx-(x+x^2y)dy=0`

Concept: Equations Reducible to Exact Form by Using Integrating Factors

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

Concept: Exact Differential Equations

Solve `ydx+x(1-3x^2y^2)dy=0`

Concept: Equations Reducible to Exact Form by Using Integrating Factors

Prove that for an astroid ` x^(2/3) +y2/3= a^(2/3)` the line 𝜽=𝝅/𝟔 Divide the arc in the first quadrant in a ratio 1:3.

Concept: Exact Differential Equations

Solve `x^2 (d^2y)/dx^2+3x dy/dx+3y =(log x.cos (log x))/x`

Concept: Equations Reducible to Exact Form by Using Integrating Factors

Evaluate `int_0^1 x^5 sin ^-1 x dx`and find the value of β `(9/2,1/2)`

Concept: Exact Differential Equations

Evaluate `int_0^6 dx/(1+3x)`by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule.

Concept: Simple Application of Differential Equation of First Order and First Degree to Electrical and Mechanical Engineering Problem

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

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Show that `int_0^1(x^a-1)/logx dx=log(a+1)`

Concept: Method of Variation of Parameters

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 by variation of parameter method `(d^2y)/(dx^2)+3(dy)/(dx)+2y=e^(e^x)`.

Concept: Method of Variation of Parameters

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

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

Given `int_0^x 1/(x^2+a^2) dx=1/atan^(-1)(x/a)`using DUIS find the value of `int_0^x 1/(x^2+a^2) `

Concept: Method of Variation of Parameters

Evaluate `int_0^∞ 3^(-4x^2) dx`

Concept: Legendre’S Differential Equation

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