MTE-08 Solved Assignment January 2026 | अवकल समीकरण | IGNOU B.SC.

Q: State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.

The following statements regarding differential equations are analyzed for their truthfulness, with justifications provided through proofs or counter-examples, adhering to the principles outlined in the MTE-08 course material.

Q: Solve the following problems:

Answer available — download the full PDF for complete details.

Q: Solve the following problems:

Answer available — download the full PDF for complete details.

Q1. State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.

(i)) The initial value problem dy/dx = x² + y², y(0) = 0 has a unique solution in some interval of the form - h<x<h. (75 words)
(ii)) The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the x-axis is x² + 2y² = c2. (75 words)
(iii)) The normal form of the differential equation y'' - 4xy' + (4x² -1) y = -3ex² sin 2x is d²v/dx² + v =-3sin 2x, where v=ye⁻x². (75 words)
(iv)) The solution of the pde ∂z/∂x + ∂z/∂y =z² is z = −[y + f (x - y)]. (75 words)
(v)) The pde uxx + x² uxy - (x²/4 + 2) uyy = 0 is hyperbolic in the entire xy-plane. (75 words)

Key Points:

Picard's theorem guarantees unique ODE solution if f and ∂f/∂y are continuous.
Orthogonal trajectories are found by replacing dy/dx with -dx/dy.
Normal form v''+Iv=S is derived by substituting y=ve^(∫-P/2 dx) into y''+Py'+Qy=R.
Method of characteristics solves P ∂z/∂x + Q ∂z/∂y = R using integral curves.

Answer: The following statements regarding differential equations are analyzed for their truthfulness, with justifications provided through proofs or counter-examples, adhering to the principles outlined in the MTE-08 course material.

Q2. Solve the following problems:

(a)) Solve dy/dx + xy = y² ex²/2 sin x. (120 words)
(b)) Write the ordinary differential equation y dx + (xy + x - 3y) dy = 0 in the linear form, and hence find its solution. (120 words)
(c)) Given that y₁(x) = x⁻¹ is one solution of the differential equation 2x²y'' + 3xy' - y = 0, x > 0, find a second linearly independent solution of the equation. (160 words)

Key Points:

Bernoulli's equation (dy/dx + P(x)y = Q(x)yⁿ) is transformed into a linear ODE using substitution v = y^(1-n).
Linear first-order ODEs (dy/dx + P(x)y = Q(x)) are solved by multiplying with an integrating factor e^(∫P(x)dx).
Integration by parts is a key technique for evaluating integrals encountered during ODE solutions.
The reduction of order method finds a second solution (y₂ = vy₁) for a second-order linear ODE if one solution (y₁) is known.

Answer: This response comprehensively addresses the three sub-questions concerning differential equations, applying various standard techniques taught in MTE-08. Sub-question (a) involves solving a Bernoulli's equation, which is first transformed into a linear first-order equation using a suitable substitution. Sub-question (b) requires rewriting a non-linear differential equation into a linear form in terms of dx/dy and then solving it using an integrating factor. Finally, sub-question (c) utilizes the...

Q3. Solve the following problems:

(a)) Solve, using the method of variation of parameters d²y/dx² - 2/x² y = 1/(1+ex) (200 words)
(b)) Solve the following equation by changing the independent variable (1+x²)² y''+2x(1+x²) y' + 4y = 0. (200 words)

Key Points:

Variation of parameters solves non-homogeneous ODEs (y''+P(x)y'+Q(x)y=R(x)).
Method steps: find y_h, calculate Wronskian W(y₁, y₂), then y_p = -y₁∫(y₂R/W)dx + y₂∫(y₁R/W)dx.
Changing independent variable transforms ODE for simpler solution, often to constant coefficients.
For (1+x²)²y''+2x(1+x²)y'+4y=0, substitution z=arctan(x) leads to d²y/dz²+4y=0.

Answer: This response addresses two problems from MTE-08 Differential Equations, focusing on solving second-order linear differential equations using specific methods. Sub-question (a) demonstrates the method of variation of parameters for a non-homogeneous equation, while sub-question (b) employs the technique of changing the independent variable to solve a homogeneous equation. Both solutions follow the systematic procedures taught in the course, providing step-by-step calculations and explanations ta...

Q4. Solve the following problems:

(a)) Find the integrating factor of the differential equation (6xy-3y² + 2y) dx + 2(x - y) dy = 0 and hence solve it. (120 words)
(b)) Solve the equation x² d²y/dx² + x dy/dx + y = xm, for all positive integer values of m. (120 words)
(c)) Solve the following IVP d²y/dx² + dy/dx - 2y = -6sin 2x - 18cos 2x y(0) = 2, y'(0) = 2. (160 words)

Key Points:

Integrating factor for M dx + N dy = 0 is e^∫f(x)dx if (∂M/∂y - ∂N/∂x)/N = f(x).
Cauchy-Euler equations (x²y''+xy'+y=f(x)) are solved by substituting x = e^t.
For Cauchy-Euler, x=e^t transforms the equation into a linear DE with constant coefficients (D_t² + 1)y = e^(mt).
Non-homogeneous linear DEs with constant coefficients are solved by y = yc + yp.

Answer:

Q5. Solve the following problems:

(a)) Solve: d²y/dx² - 2 tan x dy/dx + 5y = ex sec x. (160 words)
(b)) Find the charge on the capacitor in an RLC circuit at t = 0.01 sec. when L = 0.05 Henry, R = 2 ohms, C=0.01 Farad. E(t) = 0, q(0) = 5 Columbus and i(0) = 0. (120 words)
(c)) Solve: x dy/dx + y ln y = x y ex. (120 words)

Key Points:

Substitution `y = z sec x` can transform specific variable-coefficient ODEs into constant-coefficient ones.
RLC circuits are modeled by `L d²q/dt² + R dq/dt + (1/C)q = E(t)`, a second-order linear ODE.
Complex characteristic roots `m = α ± iβ` lead to solutions of the form `e^(αt)(C1 cos(βt) + C2 sin(βt))`.
Bernoulli-type equations like `(1/y)y' + P(x)ln(y) = Q(x)` can be linearized via `v = ln y`.

Answer: This response comprehensively addresses three differential equation problems, applying various techniques from the IGNOU MTE-08 course. Problem (a) involves a transformation to solve a second-order linear ODE with variable coefficients. Problem (b) utilizes a second-order linear homogeneous ODE to model an RLC circuit and determine capacitor charge. Problem (c) requires a substitution to convert a non-linear first-order ODE into a linear one, followed by the integrating factor method and integra...

Q6. Solve the following problems:

(a)(i)) d²y/dx² + (dy/dx + 1)² y = sin(x/2) + ex + x. (120 words)
(a)(ii)) 2x²(d²y/dx²) + (dy/dx)² + 4y² = x² + 2xy (dy/dx) (160 words)
(b)) The differential equation of a damped vibrating system under the action of an external periodic force is: d²x/dt² + 2m₀ dx/dt + n²x = a cos pt Show that, if n > m > 0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions. (120 words)

Key Points:

Non-linear ODEs: Identified by terms like (dy/dx)² or products of dependent variable/derivatives.
Non-solvable Analytically: Complex non-linear ODEs generally lack elementary analytical solutions; numerical methods are used.
Damped Oscillations (CF): Occur when characteristic equation roots are complex with a negative real part (e.g., e^(-m₀t)cos(ωt)).
Exponential Decay: The e^(-m₀t) term (m₀ > 0) causes vibration amplitude to decrease over time.

Answer: The provided problems encompass various types of differential equations, from complex non-linear forms to standard second-order linear equations crucial for modeling physical systems. A comprehensive approach involves identifying the equation type, applying appropriate solution methodologies, and interpreting the results within the context of the MTE-08 course. For non-linear equations, the primary 'solution' often involves classification and discussing the limitations of analytical methods, whi...

Q7. Solve the following problems:

(a)) Verify that the Pfaffian differential equation yz dx + (x²y – zx) dy + (x²z−xy) dz = 0 is integrable and hence find its integral. (120 words)
(b)) Solve the following equation by Jacobi's method (∂u/∂x)² - x²(∂u/∂y)² - (∂u/∂z)² = 0. (120 words)
(c)) Show that 2z = (ax + y)² + b, where a, b are arbitrary constants is a complete integral of px + qy - q² = 0. (160 words)

Key Points:

Pfaffian equation integrability requires P(∂Q/∂z - ∂R/∂y) + Q(∂R/∂x - ∂P/∂z) + R(∂P/∂y - ∂Q/∂x) = 0.
The integral of a Pfaffian equation can be found by exact differential grouping or integrating factors.
Jacobi's method for F(x, y, z, p, q, r)=0 seeks a complete integral by setting partial derivatives (e.g., p, r) to constants.
The total differential du = p dx + q dy + r dz is integrated to find the complete integral u.

Answer:

Q8. Solve the following problems:

(a)(i)) x²p + y²q = (x + y) z. (120 words)
(a)(ii)) √p - √q + 3x = 0. (120 words)
(b)) Find the equation of the integral surface of the differential equation (x² - yz) p + (y² – zx) q = z² – xy which passes through the line x = 1, y = 0. (160 words)

Key Points:

Lagrange's PDE (Pp+Qq=R) solved using auxiliary equations dx/P=dy/Q=dz/R.
Method of multipliers involves choosing factors (l,m,n) such that ldx+mdy+ndz / (lP+mQ+nR) is integrable or its denominator is zero.
General solution of Lagrange's PDE is Φ(u,v)=0, where u and v are two independent integrals.
Non-linear PDE of type f(x,p)=g(q) can be solved by setting each side to an arbitrary constant 'a'.

Answer: This response provides detailed solutions to three partial differential equation problems, covering both linear Lagrange's equations and a specific type of non-linear first-order PDE. The solutions demonstrate the application of the method of auxiliary equations and the method of multipliers for Lagrange's equations, along with the separation of variables technique for certain non-linear forms. For the integral surface problem, the procedure involves finding the general solution and then applyin...

Q9. Solve the following problems:

(a)) Using the method of separation of variables, solve uxt = e⁻ᵗ cosx when u(x, 0) = 0, ∂u/∂t (0, t) = 0. (120 words)
(b)) Find the temperature in a bar of length l with both ends insulated and with initial temperature in the rod being sin(πx/l). (280 words)

Key Points:

Separation of variables technique assumes a product solution, reducing a PDE to ODEs.
Initial conditions determine arbitrary functions/constants after integration of the PDE.
Boundary conditions are crucial for finding specific solutions and series coefficients.
The heat equation `∂u/∂t = k ∂²u/∂x²` models temperature distribution in a rod.

Answer:

Q10. Solve the following problems:

(a)(i)) [D³ – DD´² – D² + DD'] z = 0. (80 words)
(a)(ii)) [D⁴ – D´⁴ – 2D² D´² ] z = 0. (80 words)
(a)(iii)) [D² – 2DD' + D'²] z=12xy. (120 words)
(b)) Show that the wave equation a²uxx = utt can be reduced to the form uξη = 0 by the change of variable ξ = x − at, η = x + at. (120 words)

Key Points:

Homogeneous linear PDE solution uses auxiliary equation (D=m, D'=1) to find characteristic roots.
Repeated roots (m=c) in CF are handled by multiplying terms by `x` (e.g., `φ(y+cx) + xφ₂(y+cx)`).
Non-homogeneous linear PDE general solution is sum of Complementary Function (CF) and Particular Integral (PI).
Particular Integral for polynomial RHS uses operator expansion `1/f(D,D')` (e.g., binomial series).

Answer:

MTE-08 Solved Assignment — January 2026 / July 2026

अवकल समीकरण | IGNOU Bachelor of Science (B.SC.)

MTE-08—January 2026 / July 2026

अवकल समीकरण

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MTE-08 Assignment Questions — January 2026

Q1. State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.

Q2. Solve the following problems:

Q3. Solve the following problems:

Q4. Solve the following problems:

Q5. Solve the following problems:

Q6. Solve the following problems:

Q7. Solve the following problems:

Q8. Solve the following problems:

Q9. Solve the following problems:

Q10. Solve the following problems:

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