BE Instrumentation Engineering Semester 2 (FE First Year) - University of Mumbai Important Questions for Applied Mathematics 2

Evaluate ${\displaystyle {\int }_0^1{\int }_0^{x2}\frac{y}{ex}dy dx}$ 

Solve ${\displaystyle (D^2+2)y=e^x\cos x+x^2{e}^{3x}}$

Evaluate ${\displaystyle {\int }_0^{\infty }{5}^{-4x^2}dx}$

Solve : ${\displaystyle (1+\log x.y)dx+(1+\frac{x}{y})}$dy=0

Solve  ${\displaystyle xy(1+x{y}^2)\frac{dy}{dx}=1}$

Solve ${\displaystyle (y-x{y}^2)dx-(x+x^{2}y)dy=0}$

Solve the ODE ${\displaystyle (y+\frac{1}{3}{y}^3+\frac{1}{2}{x}^2)dx+(x+x{y}^2)dy=0}$

Solve ${\displaystyle ydx+x(1-3x^2{y}^2)dy=0}$

Prove that for an astroid  ${\displaystyle x^{\frac{2}{3}}+y\frac{2}{3}=a^{\frac{2}{3}}}$ the line 𝜽=𝝅/𝟔 Divide the arc in the first quadrant in a ratio 1:3. 

Solve ${\displaystyle x^2\frac{d^{2}y}{{dx}^2}+3x\frac{dy}{dx}+3y=\frac{\log x.\cos \left(\log x\right)}{x}}$

Evaluate ${\displaystyle {\int }_0^1{x}^5{\sin }^{-1}xdx}$and find the value of β ${\displaystyle (\frac{9}{2},\frac{1}{2})}$ 

Evaluate ${\displaystyle {\int }_0^6\frac{dx}{1+3x}}$by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule. 

Evaluate ${\displaystyle \frac{d^{4}y}{{dx}^4}+2\frac{d^{2}y}{{dx}^2}+y=0}$

Show that ${\displaystyle {\int }_0^1\frac{x^a-1}{\log x}dx=\log (a+1)}$

Evaluate ${\displaystyle {(2x+1)}^2\frac{d^{2}y}{{dx}^2}-2(2x+1)\frac{dy}{dx}-12y=6x}$

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 ${\displaystyle \frac{di}{dt}+Ri=E.}$

Solve by variation of parameter method ${\displaystyle \frac{d^{2}y}{{dx}^2}+3\frac{dy}{dx}+2y=e^{e^x}}$.

Solve ${\displaystyle {(D^3+1)}^{2}y=0}$

Given ${\displaystyle {\int }_0^x\frac{1}{x^2+a^2}dx=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)}$using DUIS find the value of ${\displaystyle {\int }_0^x\frac{1}{x^2+a^2}}$

Evaluate ${\displaystyle {\int }_0^{\infty }{3}^{-4x^2}dx}$