# Ordinary Differential Equation: ODE GATE Solutions 2021-2018

ODE GATE Questions are very limited. Yearly GATE (Graduate Aptitude Test in Engineering) contains a minimum of two or three questions hardly of the topic ODE. So here are the ODE GATE Solutions from the years 2021, 2020, 2019, and 2018. This our first post on chapterwise gate mathematics solved papers pdf on the topic ODE. In this article, we have 2018-2021 gate mathematics questions with solutions. Here are 2021 ODE **gate mathematics questions with solutions**.

## Ordinary Differential Equation (ODE) GATE 2021-2018 Solution

Here’s the syllabus of Ordinary Differential Equation in GATE Mathematics:

- First-order ordinary differential equations,
- Existence and uniqueness theorems for initial value problems
- Systems of linear first-order ordinary differential equations
- Linear ordinary differential equations of higher order with constant coefficients
- Linear second-order ordinary differential equations with variable coefficients
- Method of Laplace transforms for solving ordinary differential equations
- Series solutions (power series, Frobenius method)
- Legendre and Bessel functions and their orthogonal properties.

There are also topics in the GATE exam outside the syllabus. Just like there was a question in 2018 GATE of the topic regular singularity of ODE.

Q.1 The eigenvalues of the boundary value problem:

\frac{d^{2} y}{d x^{2}}+\lambda y=0, \quad x \in(0, \pi), \quad \lambda>0; y(0)=0, \quad y(\pi)-\frac{d y}{d x}(\pi)=0. Then

(A) \lambda=(n \pi)^{2}, \quad n=1,2,3, \ldots

(B) \lambda=n^{2}, \quad n=1,2,3, \ldots

(C) \lambda=k_{n}^{2}, where k_{n}, n=1,2,3, \ldots are the roots of k-\tan (k \pi)=0

(D) \lambda=k_{n}^{2}, where k_{n}, n=1,2,3, \ldots are the roots of k+\tan (k \pi)=0

Solution:

Q.2 Let y(x) be the solution of the following initial value problem:

x^{2} \frac{d^{2} y}{d x^{2}}-4 x \frac{d y}{d x}+6 y=0; x>0, y(2)=0, \frac{d y}{d x}(2)=4.

Then y(4) =.

Solution:

Q5. The initial value problem y^{\prime}=y^{\frac{3}{5}}, y(0)=b has

(A) a unique solution if b = 0

(B) no solution if b = 1

(C) infinitely many solutions if b = 2

(D) a unique solution if b = 1.

Solution:

Q.4 Let u be a solution of the differential equation y’ + xy = 0 and let \phi=u \psi be a solution of the differential cquation y^{\prime \prime}+2 x y^{\prime}+\left(x^{2}+2\right) y=0 satisfying \phi(0) = 1 and \phi{\prime}(0) = 0. Then \phi(x) is

Solution:

Q.5 Consider the boundary value problem (BVP): \frac{d^{2} y}{d x^{2}}+\alpha y(x)=0, \alpha \in \mathbb{R} (the set of all real numbers), with the boundary conditions) y(0)=0, y(\pi)=k (x is a non-zero real number).

Then which one of the following statements is TRUE?

(A) For \alpha=1, the BVP has infinitely many solutions

(B) For \alpha=1, the BVP has a unique solution

(C) For \alpha=-1, k<0, the BVP has a solution y(x) such that y(x)>0; for all x \in (0,\pi)

(D) For \alpha=-1, k>0, the BVP has a solution y(x) such that y(x)>0; for all x \in (0,\pi)

Solution:

Q.6 The general solution of the differential equation: xy'=y+\sqrt{x^{2}+y^{2}} \text { for } x > 0 \text {. }

is given by (with an arbitrary positive constant k)

(A) k y^{2}=x+\sqrt{x^{2}+y^{2}}

(B) k x^{2}=x+\sqrt{x^{2}+y^{2}}

(C) k x^{2}=y+\sqrt{x^{2}+y^{2}}

(D) k y^{2}=y+\sqrt{x^{2}+y^{2}}

Solution:

Q.7 If y_{1}(x)=e^{-x^{2}} is a solution of the differential equation x y^{\prime \prime}+\alpha y^{\prime}+\beta x^{3} y=0 for some real numbers \alpha and \beta , then \alpha \beta =

Solution:

Hope you are helped on ODE GATE Solutions from the years 2021, 2020, 2019, and 2018. If we missed any questions of any of the GATE Paper of ODE please mail us at: [email protected]