Exam 36: Solving Differential Equations by the Laplace Transform and by Numerical Methods

Solve by the Runge-Kutta method. Find the value of $y(3)$ . Take a step size of 0.1 unit. $y^{\prime \prime}+y^{\prime}+4 x y=5, y^{\prime}(2,0)=1$

(Short Answer)

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Question 41

Solve the differential equation $x y^{\prime}-y=1$ , with initial condition $y(2)=4$ and step size 0.1 , using the modified Euler's method. Find $y(3)$ to three decimal places.

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Question 42

The current in an RLC circuit satisfies the equation $0.75 i^{\prime \prime}+450 i^{\prime}+9000 i=150$ . If $i$ and $i^{\prime}$ are zero at $t=0$ , find the equation for the current.

(Short Answer)

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Question 43

In an RL circuit, $R=25 \Omega$ and $L=0.25 \mathrm{H}$ . The circuit is connected to an $\mathrm{AC}$ source of $50 \sin 100 \mathrm{t} \mathrm{V}$ at $t=0$ . If $i$ is zero at $t=0$ , find the current.

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Question 44

Find the inverse transform of $\frac{1}{s^{2}+4 s+13}$ .

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Question 45

Use any numerical method to solve the differential equation: $y^{\prime}+x \sin y=x ; y(1)=2$ . Find $y(2)$ to two decimal places.

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Question 46

Use any numerical method to solve the differential equation: $2 y^{\prime}+4 y^{2}=5 x$ . If $y(0)=2$ , find $y(1)$ to three decimal places.

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Question 47

Use any numerical method to solve the following differential equation: $y^{\prime \prime}+x^{2}+y y^{\prime}=10 ; y^{\prime}(1,7)=3$ . Find $y(2)$ to three decimal places.

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Question 48

Use any numerical method to solve the differential equation: $y^{\prime}+x \sin y=x ; y(0)=5$ . Find $y(1)$ to three decimal places.

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Question 49

Find the Laplace transform of the expression and substitute in the initial conditions: $2 y^{\prime \prime}+3 y^{\prime}+y$ ; $y^{\prime}(0,1)=2$

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Question 50

Find the Laplace transform of the function $f(t)=2 t-6$ .

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Question 51

Solve the differential equation by the Laplace transform: $y^{\prime \prime}-4 y=e^{2 t} ; y^{\prime}(0,0)=5$

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Question 52

Use any numerical method to solve the following differential equation: $e^{\mathrm{x}} \mathrm{y}^{\prime}-x^{2} y-y=4 ; y(1)=7$ . Find $y(2)$ to two decimal places.

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Question 53

Find the Laplace transform of $2 y^{\prime \prime}+5 y^{\prime}-3 y$ and substitute the initial condition $y^{\prime}(0,0)=0$ .

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Question 54

Find the inverse of the transform: $\frac{s+1}{s^{3}+s}$

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Question 55

Solve by the Runge-Kutta method. Find the value of $y$ at $x=2$ . Take a step size of 0.1 unit. $y^{\prime \prime}+y^{\prime}+2 x y=8, y^{\prime}(1,2)=4$

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Question 56

Exam 36: Solving Differential Equations by the Laplace Transform and by Numerical Methods

Solve by the Runge-Kutta method. Find the value of y(3)y(3)y(3) . Take a step size of 0.1 unit. y′′+y′+4xy=5,y′(2,0)=1y^{\prime \prime}+y^{\prime}+4 x y=5, y^{\prime}(2,0)=1y′′+y′+4xy=5,y′(2,0)=1

Solve the differential equation xy′−y=1x y^{\prime}-y=1xy′−y=1 , with initial condition y(2)=4y(2)=4y(2)=4 and step size 0.1 , using the modified Euler's method. Find y(3)y(3)y(3) to three decimal places.

The current in an RLC circuit satisfies the equation 0.75i′′+450i′+9000i=1500.75 i^{\prime \prime}+450 i^{\prime}+9000 i=1500.75i′′+450i′+9000i=150 . If iii and i′i^{\prime}i′ are zero at t=0t=0t=0 , find the equation for the current.

In an RL circuit, R=25ΩR=25 \OmegaR=25Ω and L=0.25HL=0.25 \mathrm{H}L=0.25H . The circuit is connected to an AC\mathrm{AC}AC source of 50sin⁡100tV50 \sin 100 \mathrm{t} \mathrm{V}50sin100tV at t=0t=0t=0 . If iii is zero at t=0t=0t=0 , find the current.

Find the inverse transform of 1s2+4s+13\frac{1}{s^{2}+4 s+13}s2+4s+131​ .

Use any numerical method to solve the differential equation: y′+xsin⁡y=x;y(1)=2y^{\prime}+x \sin y=x ; y(1)=2y′+xsiny=x;y(1)=2 . Find y(2)y(2)y(2) to two decimal places.

Use any numerical method to solve the differential equation: 2y′+4y2=5x2 y^{\prime}+4 y^{2}=5 x2y′+4y2=5x . If y(0)=2y(0)=2y(0)=2 , find y(1)y(1)y(1) to three decimal places.

Use any numerical method to solve the following differential equation: y′′+x2+yy′=10;y′(1,7)=3y^{\prime \prime}+x^{2}+y y^{\prime}=10 ; y^{\prime}(1,7)=3y′′+x2+yy′=10;y′(1,7)=3 . Find y(2)y(2)y(2) to three decimal places.

Use any numerical method to solve the differential equation: y′+xsin⁡y=x;y(0)=5y^{\prime}+x \sin y=x ; y(0)=5y′+xsiny=x;y(0)=5 . Find y(1)y(1)y(1) to three decimal places.

Find the Laplace transform of the expression and substitute in the initial conditions: 2y′′+3y′+y2 y^{\prime \prime}+3 y^{\prime}+y2y′′+3y′+y ; y′(0,1)=2y^{\prime}(0,1)=2y′(0,1)=2

Find the Laplace transform of the function f(t)=2t−6f(t)=2 t-6f(t)=2t−6 .

Solve the differential equation by the Laplace transform: y′′−4y=e2t;y′(0,0)=5y^{\prime \prime}-4 y=e^{2 t} ; y^{\prime}(0,0)=5y′′−4y=e2t;y′(0,0)=5

Use any numerical method to solve the following differential equation: exy′−x2y−y=4;y(1)=7e^{\mathrm{x}} \mathrm{y}^{\prime}-x^{2} y-y=4 ; y(1)=7exy′−x2y−y=4;y(1)=7 . Find y(2)y(2)y(2) to two decimal places.

Find the Laplace transform of 2y′′+5y′−3y2 y^{\prime \prime}+5 y^{\prime}-3 y2y′′+5y′−3y and substitute the initial condition y′(0,0)=0y^{\prime}(0,0)=0y′(0,0)=0 .

Find the inverse of the transform: s+1s3+s\frac{s+1}{s^{3}+s}s3+ss+1​

Solve by the Runge-Kutta method. Find the value of yyy at x=2x=2x=2 . Take a step size of 0.1 unit. y′′+y′+2xy=8,y′(1,2)=4y^{\prime \prime}+y^{\prime}+2 x y=8, y^{\prime}(1,2)=4y′′+y′+2xy=8,y′(1,2)=4

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Solve by the Runge-Kutta method. Find the value of $y(3)$ . Take a step size of 0.1 unit. $y^{\prime \prime}+y^{\prime}+4 x y=5, y^{\prime}(2,0)=1$

Solve the differential equation $x y^{\prime}-y=1$ , with initial condition $y(2)=4$ and step size 0.1 , using the modified Euler's method. Find $y(3)$ to three decimal places.

The current in an RLC circuit satisfies the equation $0.75 i^{\prime \prime}+450 i^{\prime}+9000 i=150$ . If $i$ and $i^{\prime}$ are zero at $t=0$ , find the equation for the current.

In an RL circuit, $R=25 \Omega$ and $L=0.25 \mathrm{H}$ . The circuit is connected to an $\mathrm{AC}$ source of $50 \sin 100 \mathrm{t} \mathrm{V}$ at $t=0$ . If $i$ is zero at $t=0$ , find the current.

Find the inverse transform of $\frac{1}{s^{2}+4 s+13}$ .

Use any numerical method to solve the differential equation: $y^{\prime}+x \sin y=x ; y(1)=2$ . Find $y(2)$ to two decimal places.

Use any numerical method to solve the differential equation: $2 y^{\prime}+4 y^{2}=5 x$ . If $y(0)=2$ , find $y(1)$ to three decimal places.

Use any numerical method to solve the following differential equation: $y^{\prime \prime}+x^{2}+y y^{\prime}=10 ; y^{\prime}(1,7)=3$ . Find $y(2)$ to three decimal places.

Use any numerical method to solve the differential equation: $y^{\prime}+x \sin y=x ; y(0)=5$ . Find $y(1)$ to three decimal places.

Find the Laplace transform of the expression and substitute in the initial conditions: $2 y^{\prime \prime}+3 y^{\prime}+y$ ; $y^{\prime}(0,1)=2$

Find the Laplace transform of the function $f(t)=2 t-6$ .

Solve the differential equation by the Laplace transform: $y^{\prime \prime}-4 y=e^{2 t} ; y^{\prime}(0,0)=5$

Use any numerical method to solve the following differential equation: $e^{\mathrm{x}} \mathrm{y}^{\prime}-x^{2} y-y=4 ; y(1)=7$ . Find $y(2)$ to two decimal places.

Find the Laplace transform of $2 y^{\prime \prime}+5 y^{\prime}-3 y$ and substitute the initial condition $y^{\prime}(0,0)=0$ .

Find the inverse of the transform: $\frac{s+1}{s^{3}+s}$

Solve by the Runge-Kutta method. Find the value of $y$ at $x=2$ . Take a step size of 0.1 unit. $y^{\prime \prime}+y^{\prime}+2 x y=8, y^{\prime}(1,2)=4$